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Fri, 17 May 2019 07:56:21 -0700Fri, 17 May 2019 07:56:21 -0700Jekyll v3.7.2Fast Fourier Transforms<p><em>Trigger warning: specialized mathematical topic</em></p>
<p><em>Special thanks to Karl Floersch for feedback</em></p>
<p>One of the more interesting algorithms in number theory is the Fast Fourier transform (FFT). FFTs are a key building block in many algorithms, including <a href="http://www.math.clemson.edu/~sgao/papers/GM10.pdf">extremely fast multiplication of large numbers</a>, multiplication of polynomials, and extremely fast generation and recovery of <a href="https://blog.ethereum.org/2014/08/16/secret-sharing-erasure-coding-guide-aspiring-dropbox-decentralizer">erasure codes</a>. Erasure codes in particular are highly versatile; in addition to their basic use cases in fault-tolerant data storage and recovery, erasure codes also have more advanced use cases such as <a href="https://arxiv.org/pdf/1809.09044">securing data availability in scalable blockchains</a> and <a href="https://vitalik.ca/general/2017/11/09/starks_part_1.html">STARKs</a>. This article will go into what fast Fourier transforms are, and how some of the simpler algorithms for computing them work.</p>
<h3>Background</h3>
<p>The original <a href="https://en.wikipedia.org/wiki/Fourier_transform">Fourier transform</a> is a mathematical operation that is often described as converting data between the "frequency domain" and the "time domain". What this means more precisely is that if you have a piece of data, then running the algorithm would come up with a collection of sine waves with different frequencies and amplitudes that, if you added them together, would approximate the original data. Fourier transforms can be used for such wonderful things as <a href="https://twitter.com/johncarlosbaez/status/1094671748501405696">expressing square orbits through epicycles</a> and <a href="https://en.wikipedia.org/wiki/Fourier_transform">deriving a set of equations that can draw an elephant</a>:</p>
<p><center><table><tr><td>
<img src="http://vitalik.ca/files/elephant1.png" /><br />
<img src="http://vitalik.ca/files/elephant3.png" />
</td><td>
<img src="http://vitalik.ca/files/elephant2.png" width="400px" />
</td></tr></table><br />
<small><i>Ok fine, Fourier transforms also have really important applications in signal processing, quantum mechanics, and other areas, and help make significant parts of the global economy happen. But come on, elephants are cooler.</i></small>
</center><br /></p>
<p>Running the Fourier transform algorithm in the "inverse" direction would simply take the sine waves and add them together and compute the resulting values at as many points as you wanted to sample. </p>
<p>The kind of Fourier transform we'll be talking about in this post is a similar algorithm, except instead of being a <em>continuous</em> Fourier transform over <em>real or complex numbers</em>, it's a <em><strong>discrete Fourier transform</strong></em> over <em>finite fields</em> (see the "A Modular Math Interlude" section <a href="https://vitalik.ca/general/2017/11/22/starks_part_2.html">here</a> for a refresher on what finite fields are). Instead of talking about converting between "frequency domain" and "time domain", here we'll talk about two different operations: <em>multi-point polynomial evaluation</em> (evaluating a degree < N polynomial at N different points) and its inverse, <em>polynomial interpolation</em> (given the evaluations of a degree < N polynomial at N different points, recovering the polynomial). For example, if we are operating in the prime field with modulus 5, then the polynomial <code>y = x² + 3</code> (for convenience we can write the coefficients in increasing order: <code>[3,0,1]</code>) evaluated at the points <code>[0,1,2]</code> gives the values <code>[3,4,2]</code> (not <code>[3, 4, 7]</code> because we're operating in a finite field where the numbers wrap around at 5), and we can actually take the evaluations <code>[3,4,2]</code> and the coordinates they were evaluated at (<code>[0,1,2]</code>) to recover the original polynomial <code>[3,0,1]</code>.</p>
<p>There are algorithms for both multi-point evaluation and interpolation that can do either operation in O(N<sup>2</sup>) time. Multi-point evaluation is simple: just separately evaluate the polynomial at each point. Here's python code for doing that:</p>
<pre>
def eval_poly_at(self, poly, x, modulus):
y = 0
power_of_x = 1
for coefficient in poly:
y += power_of_x * coefficient
power_of_x *= x
return y % modulus
</pre>
<p>The algorithm runs a loop going through every coefficient and does one thing for each coefficient, so it runs in O(N) time. Multi-point evaluation involves doing this evaluation at N different points, so the total run time is O(N<sup>2</sup>).</p>
<p>Lagrange interpolation is more complicated (search for "Lagrange interpolation" <a href="https://blog.ethereum.org/2014/08/16/secret-sharing-erasure-coding-guide-aspiring-dropbox-decentralizer/">here</a> for a more detailed explanation). The key building block of the basic strategy is that for any domain <code>D</code> and point <code>x</code>, we can construct a polynomial that returns 1 for <code>x</code> and 0 for any value in <code>D</code> other than <code>x</code>. For example, if <code>D = [1,2,3,4]</code> and <code>x = 1</code>, the polynomial is:</p>
<p><center>
<img src="https://vitalik.ca/files/CodeCogsEqn-19.gif" /><br />
</center><br /></p>
<p>You can mentally plug in 1, 2, 3 and 4 to the above expression and verify that it returns 1 for x=1 and 0 in the other three cases.</p>
<p>We can recover the polynomial that gives any desired set of outputs on the given domain by multiplying and adding these polynomials. If we call the above polynomial <code>P_1</code>, and the equivalent ones for <code>x=2</code>, <code>x=3</code>, <code>x=4</code>, <code>P_2</code>, <code>P_3</code> and <code>P_4</code>, then the polynomial that returns <code>[3,1,4,1]</code> on the domain <code>[1,2,3,4]</code> is simply <code>3 * P_1 + P_2 + 4 * P_3 + P_4</code>. Computing the <code>P_i</code> polynomials takes O(N<sup>2</sup>) time (you first construct the polynomial that returns to 0 on the entire domain, which takes O(N<sup>2</sup>) time, then separately divide it by <code>(x - x_i)</code> for each <code>x_i</code>), and computing the linear combination takes another O(N<sup>2</sup>) time, so it's O(N<sup>2</sup>) runtime total.</p>
<p>What Fast Fourier transforms let us do, is make both multi-point evaluation and interpolation much faster.</p>
<h3>Fast Fourier Transforms</h3>
<p>There is a price you have to pay for using this much faster algorithm, which is that you cannot choose any arbitrary field and any arbitrary domain. Whereas with Lagrange interpolation, you could choose whatever x coordinates and y coordinates you wanted, and whatever field you wanted (you could even do it over plain old real numbers), and you could get a polynomial that passes through them., with an FFT, you have to use a finite field, and the domain must be a <em>multiplicative subgroup</em> of the field (that is, a list of powers of some "generator" value). For example, you could use the finite field of integers modulo 337, and for the domain use <code>[1, 85, 148, 111, 336, 252, 189, 226]</code> (that's the powers of 85 in the field, eg. <code>85³ % 337 = 111</code>; it stops at 226 because the next power of 85 cycles back to 1). Futhermore, the multiplicative subgroup must have size 2<sup>n</sup> (there's ways to make it work for numbers of the form 2<sup>m</sup> * 3<sup>n</sup> and possibly slightly higher prime powers but then it gets much more complicated and inefficient). The finite field of intergers modulo 59, for example, would not work, because there are only multiplicative subgroups of order 2, 29 and 58; 2 is too small to be interesting, and the factor 29 is far too large to be FFT-friendly. The symmetry that comes from multiplicative groups of size 2<sup>n</sup> lets us create a recursive algorithm that quite cleverly calculate the results we need from a much smaller amount of work.</p>
<p>To understand the algorithm and why it has a low runtime, it's important to understand the general concept of recursion. A recursive algorithm is an algorithm that has two cases: a "base case" where the input to the algorithm is small enough that you can give the output directly, and the "recursive case" where the required computation consists of some "glue computation" plus one or more uses of the same algorithm to smaller inputs. For example, you might have seen recursive algorithms being used for sorting lists. If you have a list (eg. <code>[1,8,7,4,5,6,3,2,9]</code>), then you can sort it using the following procedure:</p>
<ul>
<li>If the input has one element, then it's already "sorted", so you can just return the input.</li>
<li>If the input has more than one element, then separately sort the first half of the list and the second half of the list, and then merge the two sorted sub-lists (call them A and B) as follows. Maintain two counters, <code>apos</code> and <code>bpos</code>, both starting at zero, and maintain an output list, which starts empty. Until either <code>apos</code> or <code>bpos</code> is at the end of the corresponding list, check if <code>A[apos]</code> or <code>B[bpos]</code> is smaller. Whichever is smaller, add that value to the end of the output list, and increase that counter by 1. Once this is done, add the rest of whatever list has not been fully processed to the end of the output list, and return the output list.</li>
</ul>
<p>Note that the "glue" in the second procedure has runtime O(N): if each of the two sub-lists has <code>N</code> elements, then you need to run through every item in each list once, so it's O(N) computation total. So the algorithm as a whole works by taking a problem of size <code>N</code>, and breaking it up into two problems of size <code>N/2</code>, plus O(N) of "glue" execution. There is a theorem called the <a href="https://en.wikipedia.org/wiki/Master_theorem_(analysis_of_algorithms%29">Master Theorem</a> that lets us compute the total runtime of algorithms like this. It has many sub-cases, but in the case where you break up an execution of size <code>N</code> into <code>k</code> sub-cases of size <code>N/k</code> with O(N) glue (as is the case here), the result is that the execution takes time O(N * log(N)).</p>
<p><center>
<img src="http://vitalik.ca/files/sorting.png" /><br />
</center><br /></p>
<p>An FFT works in the same way. We take a problem of size <code>N</code>, break it up into two problems of size <code>N/2</code>, and do O(N) glue work to combine the smaller solutions into a bigger solution, so we get O(N * log(N)) runtime total - <em>much faster</em> than O(N<sup>2</sup>). Here is how we do it. I'll describe first how to use an FFT for multi-point evaluation (ie. for some domain <code>D</code> and polynomial <code>P</code>, calculate <code>P(x)</code> for every <code>x</code> in <code>D</code>), and it turns out that you can use the same algorithm for interpolation with a minor tweak.</p>
<p>Suppose that we have an FFT where the given domain is the powers of <code>x</code> in some field, where x<sup>2<sup>k</sup></sup> = 1 (eg. in the case we introduced above, the domain is the powers of 85 modulo 337, and 85<sup>2<sup>3</sup></sup> = 1). We have some polynomial, eg. <code>y = 6x⁷ + 2x⁶ + 9x⁵ + 5x⁴ + x³ + 4x² + x + 3</code> (we'll write it as <code>p = [3, 1, 4, 1, 5, 9, 2, 6]</code>). We want to evaluate this polynomial at each point in the domain, ie. at each of the eight powers of 85. Here is what we do. First, we break up the polynomial into two parts, which we'll call <code>evens</code> and <code>odds</code>: <code>evens = [3, 4, 5, 2]</code> and <code>odds = [1, 1, 9, 6]</code> (or <code>evens = 2x³ + 5x² + 4x + 3</code> and <code>odds = 6x³ + 9x² + x + 1</code>; yes, this is just taking the even-degree coefficients and the odd-degree coefficients). Now, we note a mathematical observation: <code>p(x) = evens(x²) + x * odds(x²)</code> and <code>p(-x) = evens(x²) - x * odds(x²)</code> (think about this for yourself and make sure you understand it before going further).</p>
<p>Here, we have a nice property: <code>evens</code> and <code>odds</code> are both polynomials half the size of <code>p</code>, and furthermore, the set of possible values of <code>x²</code> is only half the size of the original domain, because there is a two-to-one correspondence: <code>x</code> and <code>-x</code> are both part of <code>D</code> (eg. in our current domain <code>[1, 85, 148, 111, 336, 252, 189, 226]</code>, 1 and 336 are negatives of each other, as <code>336 = -1 % 337</code>, as are <code>(85, 252)</code>, <code>(148, 189)</code> and <code>(111, 226)</code>. And <code>x</code> and <code>-x</code> always both have the same square. Hence, we can use an FFT to compute the result of <code>evens(x)</code> for every <code>x</code> in the smaller domain consisting of squares of numbers in the original domain (<code>[1, 148, 336, 189]</code>), and we can do the same for odds. And voila, we've reduced a size-N problem into half-size problems.</p>
<p>The "glue" is relatively easy (and O(N) in runtime): we receive the evaluations of <code>evens</code> and <code>odds</code> as size-<code>N/2</code> lists, so we simply do <code>p[i] = evens_result[i] + domain[i] * odds_result[i]</code> and <code>p[N/2 + i] = evens_result[i] - domain[i] * odds_result[i]</code> for each index <code>i</code>.</p>
<p>Here's the full code:</p>
<pre>
def fft(vals, modulus, domain):
if len(vals) == 1:
return vals
L = fft(vals[::2], modulus, domain[::2])
R = fft(vals[1::2], modulus, domain[::2])
o = [0 for i in vals]
for i, (x, y) in enumerate(zip(L, R)):
y_times_root = y*domain[i]
o[i] = (x+y_times_root) % modulus
o[i+len(L)] = (x-y_times_root) % modulus
return o
</pre>
<p>We can try running it:</p>
<pre>
>>> fft([3,1,4,1,5,9,2,6], 337, [1, 85, 148, 111, 336, 252, 189, 226])
[31, 70, 109, 74, 334, 181, 232, 4]
</pre>
<p>And we can check the result; evaluating the polynomial at the position 85, for example, actually does give the result 70. Note that this only works if the domain is "correct"; it needs to be of the form <code>[x**i % modulus for i in range(n)]</code> where <code>x**n == 1</code>.</p>
<p>An inverse FFT is surprisingly simple:</p>
<pre>
def inverse_fft(vals, modulus, domain):
vals = fft(vals, modulus, domain)
return [x * modular_inverse(len(vals), modulus) % modulus for x in [vals[0]] + vals[1:][::-1]]
</pre>
<p>Basically, run the FFT again, but reverse the result (except the first item stays in place) and divide every value by the length of the list.</p>
<pre>
>>> domain = [1, 85, 148, 111, 336, 252, 189, 226]
>>> def modular_inverse(x, n): return pow(x, n - 2, n)
>>> values = fft([3,1,4,1,5,9,2,6], 337, domain)
>>> values
[31, 70, 109, 74, 334, 181, 232, 4]
>>> inverse_fft(values, 337, domain)
[3, 1, 4, 1, 5, 9, 2, 6]
</pre>
<p>Now, what can we use this for? Here's one fun use case: we can use FFTs to multiply numbers very quickly. Suppose we wanted to multiply 1253 by 1895. Here is what we would do. First, we would convert the problem into one that turns out to be slightly easier: multiply the <em>polynomials</em> <code>[3, 5, 2, 1]</code> by <code>[5, 9, 8, 1]</code> (that's just the digits of the two numbers in increasing order), and then convert the answer back into a number by doing a single pass to carry over tens digits. We can multiply polynomials with FFTs quickly, because it turns out that if you convert a polynomial into <em>evaluation form</em> (ie. <code>f(x)</code> for every <code>x</code> in some domain <code>D</code>), then you can multiply two polynomials simply by multiplying their evaluations. So what we'll do is take the polynomials representing our two numbers in <em>coefficient form</em>, use FFTs to convert them to evaluation form, multiply them pointwise, and convert back:</p>
<pre>
>>> p1 = [3,5,2,1,0,0,0,0]
>>> p2 = [5,9,8,1,0,0,0,0]
>>> x1 = fft(p1, 337, domain)
>>> x1
[11, 161, 256, 10, 336, 100, 83, 78]
>>> x2 = fft(p2, 337, domain)
>>> x2
[23, 43, 170, 242, 3, 313, 161, 96]
>>> x3 = [(v1 * v2) % 337 for v1, v2 in zip(x1, x2)]
>>> x3
[253, 183, 47, 61, 334, 296, 220, 74]
>>> inverse_fft(x3, 337, domain)
[15, 52, 79, 66, 30, 10, 1, 0]
</pre>
<p>This requires three FFTs (each O(N * log(N)) time) and one pointwise multiplication (O(N) time), so it takes O(N * log(N)) time altogether (technically a little bit more than O(N * log(N)), because for very big numbers you would need replace 337 with a bigger modulus and that would make multiplication harder, but close enough). This is <em>much faster</em> than schoolbook multiplication, which takes O(N<sup>2</sup>) time:</p>
<pre>
3 5 2 1
------------
5 | 15 25 10 5
9 | 27 45 18 9
8 | 24 40 16 8
1 | 3 5 2 1
---------------------
15 52 79 66 30 10 1
</pre>
<p>So now we just take the result, and carry the tens digits over (this is a "walk through the list once and do one thing at each point" algorithm so it takes O(N) time):</p>
<pre>
[15, 52, 79, 66, 30, 10, 1, 0]
[ 5, 53, 79, 66, 30, 10, 1, 0]
[ 5, 3, 84, 66, 30, 10, 1, 0]
[ 5, 3, 4, 74, 30, 10, 1, 0]
[ 5, 3, 4, 4, 37, 10, 1, 0]
[ 5, 3, 4, 4, 7, 13, 1, 0]
[ 5, 3, 4, 4, 7, 3, 2, 0]
</pre>
<p>And if we read the digits from top to bottom, we get 2374435. Let's check the answer....</p>
<pre>
>>> 1253 * 1895
2374435
</pre>
<p>Yay! It worked. In practice, on such small inputs, the difference between O(N * log(N)) and O(N<sup>2</sup>) isn't <em>that</em> large, so schoolbook multiplication is faster than this FFT-based multiplication process just because the algorithm is simpler, but on large inputs it makes a really big difference.</p>
<p>But FFTs are useful not just for multiplying numbers; as mentioned above, polynomial multiplication and multi-point evaluation are crucially important operations in implementing erasure coding, which is a very important technique for building many kinds of redundant fault-tolerant systems. If you like fault tolerance and you like efficiency, FFTs are your friend.</p>
<h3>FFTs and binary fields</h3>
<p>Prime fields are not the only kind of finite field out there. Another kind of finite field (really a special case of the more general concept of an <em>extension field</em>, which are kind of like the finite-field equivalent of complex numbers) are binary fields. In an binary field, each element is expressed as a polynomial where all of the entries are 0 or 1, eg. <code>x³ + x + 1</code>. Adding polynomials is done modulo 2, and subtraction is the same as addition (as -1 = 1 mod 2). We select some irreducible polynomial as a modulus (eg. <code>x⁴ + x + 1</code>; <code>x⁴ + 1</code> would not work because <code>x⁴ + 1</code> can be factored into <code>(x² + 1) * (x² + 1)</code> so it's not "irreducible"); multiplication is done modulo that modulus. For example, in the binary field mod <code>x⁴ + x + 1</code>, multiplying <code>x² + 1</code> by <code>x³ + 1</code> would give <code>x⁵ + x³ + x² + 1</code> if you just do the multiplication, but <code>x⁵ + x³ + x² + 1 = (x⁴ + x + 1) * x + (x³ + x + 1)</code>, so the result is the remainder <code>x³ + x + 1</code>.</p>
<p>We can express this example as a multiplication table. First multiply <code>[1, 0, 0, 1]</code> (ie. <code>x³ + 1</code>) by <code>[1, 0, 1]</code> (ie. <code>x² + 1</code>):</p>
<pre>
1 0 0 1
--------
1 | 1 0 0 1
0 | 0 0 0 0
1 | 1 0 0 1
------------
1 0 1 1 0 1
</pre>
<p>The multiplication result contains an <code>x⁵</code> term so we can subtract <code>(x⁴ + x + 1) * x</code>:</p>
<pre>
1 0 1 1 0 1
- 1 1 0 0 1 [(x⁴ + x + 1) shifted right by one to reflect being multipled by x]
------------
1 1 0 1 0 0
</pre>
<p>And we get the result, <code>[1, 1, 0, 1]</code> (or <code>x³ + x + 1</code>).</p>
<p><center>
<img src="https://vitalik.ca/files/addmult.png" style="width:600px" /><br /><br />
<small><i>Addition and multiplication tables for the binary field mod <code>x⁴ + x + 1</code>. Field elements are expressed as integers converted from binary (eg. <code>x³ + x² -> 1100 -> 12</code>)</i></small>
</center><br /></p>
<p>Binary fields are interesting for two reasons. First of all, if you want to erasure-code binary data, then binary fields are really convenient because N bytes of data can be directly encoded as a binary field element, and any binary field elements that you generate by performing computations on it will also be N bytes long. You cannot do this with prime fields because prime fields' size is not exactly a power of two; for example, you could encode every 2 bytes as a number from 0...65536 in the prime field modulo 65537 (which is prime), but if you do an FFT on these values, then the output could contain 65536, which cannot be expressed in two bytes. Second, the fact that addition and subtraction become the same operation, and 1 + 1 = 0, create some "structure" which leads to some very interesting consequences. One particularly interesting, and useful, oddity of binary fields is the "<a href="https://en.wikipedia.org/wiki/Freshman%27s_dream">freshman's dream</a>" theorem: <code>(x+y)² = x² + y²</code> (and the same for exponents 4, 8, 16... basically any power of two).</p>
<p>But if you want to use binary fields for erasure coding, and do so efficiently, then you need to be able to do Fast Fourier transforms over binary fields. But then there is a problem: in a binary field, <em>there are no (nontrivial) multiplicative groups of order 2<sup>n</sup></em>. This is because the multiplicative groups are all order 2<sup>n</sup>-1. For example, in the binary field with modulus <code>x⁴ + x + 1</code>, if you start calculating successive powers of <code>x+1</code>, you cycle back to 1 after <em>15</em> steps - not 16. The reason is that the total number of elements in the field is 16, but one of them is zero, and you're never going to reach zero by multiplying any nonzero value by itself in a field, so the powers of <code>x+1</code> cycle through every element but zero, so the cycle length is 15, not 16. So what do we do?</p>
<p>The reason we needed the domain to have the "structure" of a multiplicative group with 2<sup>n</sup> elements before is that we needed to reduce the size of the domain by a factor of two by squaring each number in it: the domain <code>[1, 85, 148, 111, 336, 252, 189, 226]</code> gets reduced to <code>[1, 148, 336, 189]</code> because 1 is the square of both 1 and 336, 148 is the square of both 85 and 252, and so forth. But what if in a binary field there's a different way to halve the size of a domain? It turns out that there is: given a domain containing 2<sup>k</sup> values, including zero (technically the domain must be a <em><a href="https://en.wikipedia.org/wiki/Linear_subspace">subspace</a></em>), we can construct a half-sized new domain <code>D'</code> by taking <code>x * (x+k) for x in D</code> using some specific <code>k</code> in <code>D</code>. Because the original domain is a subspace, since <code>k</code> is in the domain, any <code>x</code> in the domain has a corresponding <code>x+k</code> also in the domain, and the function <code>f(x) = x * (x+k)</code> returns the same value for <code>x</code> and <code>x+k</code> so we get the same kind of two-to-one correspondence that squaring gives us.</p>
<center>
<table border="1" cellpadding="10"><tr>
<td><code>x</code></td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td><td>14</td><td>15</td>
</tr><tr>
<td><code>x * (x+1)</code></td><td>0</td><td>0</td><td>6</td><td>6</td><td>7</td><td>7</td><td>1</td><td>1</td><td>4</td><td>4</td><td>2</td><td>2</td><td>3</td><td>3</td><td>5</td><td>5</td>
</tr></table>
</center>
<p><br /></p>
<p>So now, how do we do an FFT on top of this? We'll use the same trick, converting a problem with an N-sized polynomial and N-sized domain into two problems each with an N/2-sized polynomial and N/2-sized domain, but this time using different equations. We'll convert a polynomial <code>p</code> into two polynomials <code>evens</code> and <code>odds</code> such that <code>p(x) = evens(x*(k-x)) + x * odds(x*(k-x))</code>. Note that for the <code>evens</code> and <code>odds</code> that we find, it will <em>also</em> be true that <code>p(x+k) = evens(x*(k-x)) + (x+k) * odds(x*(k-x))</code>. So we can then recursively do an FFT to <code>evens</code> and <code>odds</code> on the reduced domain <code>[x*(k-x) for x in D]</code>, and then we use these two formulas to get the answers for two "halves" of the domain, one offset by <code>k</code> from the other.</p>
<p>Converting <code>p</code> into <code>evens</code> and <code>odds</code> as described above turns out to itself be nontrivial. The "naive" algorithm for doing this is itself O(N<sup>2</sup>), but it turns out that in a binary field, we can use the fact that <code>(x²-kx)² = x⁴ - k² * x²</code>, and more generally (x<sup>2</sup>-kx)<sup>2<sup>i</sup></sup> = x<sup>2<sup>i+1</sup></sup> - k<sup>2<sup>i</sup></sup> * x<sup>2<sup>i</sup></sup>, to create yet another recursive algorithm to do this in O(N * log(N)) time.</p>
<p>And if you want to do an <em>inverse</em> FFT, to do interpolation, then you need to run the steps in the algorithm in reverse order. You can find the complete code for doing this here: <a href="https://github.com/ethereum/research/tree/master/binary_fft">https://github.com/ethereum/research/tree/master/binary_fft</a>, and a paper with details on more optimal algorithms here: <a href="http://www.math.clemson.edu/~sgao/papers/GM10.pdf">http://www.math.clemson.edu/~sgao/papers/GM10.pdf</a></p>
<p>So what do we get from all of this complexity? Well, we can try running the implementation, which features both a "naive" O(N<sup>2</sup>) multi-point evaluation and the optimized FFT-based one, and time both. Here are my results:</p>
<pre>
>>> import binary_fft as b
>>> import time, random
>>> f = b.BinaryField(1033)
>>> poly = [random.randrange(1024) for i in range(1024)]
>>> a = time.time(); x1 = b._simple_ft(f, poly); time.time() - a
0.5752472877502441
>>> a = time.time(); x2 = b.fft(f, poly, list(range(1024))); time.time() - a
0.03820443153381348
</pre>
<p>And as the size of the polynomial gets larger, the naive implementation (<code>_simple_ft</code>) gets slower much more quickly than the FFT:</p>
<pre>
>>> f = b.BinaryField(2053)
>>> poly = [random.randrange(2048) for i in range(2048)]
>>> a = time.time(); x1 = b._simple_ft(f, poly); time.time() - a
2.2243144512176514
>>> a = time.time(); x2 = b.fft(f, poly, list(range(2048))); time.time() - a
0.07896280288696289
</pre>
<p>And voila, we have an efficient, scalable way to multi-point evaluate and interpolate polynomials. If we want to use FFTs to recover erasure-coded data where we are <em>missing</em> some pieces, then algorithms for this <a href="https://ethresear.ch/t/reed-solomon-erasure-code-recovery-in-n-log-2-n-time-with-ffts/3039">also exist</a>, though they are somewhat less efficient than just doing a single FFT. Enjoy!</p>
Sun, 12 May 2019 18:03:10 -0700
https://vitalik.ca/general/2019/05/12/fft.html
https://vitalik.ca/general/2019/05/12/fft.htmlgeneralControl as Liability<p>The regulatory and legal environment around internet-based services and applications has changed considerably over the last decade. When large-scale social networking platforms first became popular in the 2000s, the general attitude toward mass data collection was essentially “why not?”. This was the age of Mark Zuckerberg <a href="https://archive.nytimes.com/www.nytimes.com/external/readwriteweb/2010/01/10/10readwriteweb-facebooks-zuckerberg-says-the-age-of-privac-82963.html">saying the age of privacy is over</a> and Eric Schmidt <a href="https://www.eff.org/deeplinks/2009/12/google-ceo-eric-schmidt-dismisses-privacy">arguing</a>, “If you have something that you don’t want anyone to know, maybe you shouldn’t be doing it in the first place.” And it made personal sense for them to argue this: every bit of data you can get about others was a potential machine learning advantage for you, every single restriction a weakness, and if something happened to that data, the costs were relatively minor. Ten years later, things are very different.</p>
<p>It is especially worth zooming in on a few particular trends.</p>
<ul>
<li><strong>Privacy</strong>. Over the last ten years, a number of privacy laws have been passed, most aggressively in Europe but also elsewhere, but the most recent is <a href="https://gdpr.eu/">the GDPR</a>. The GDPR has many parts, but among the most prominent are: (i) requirements for explicit consent, (ii) requirement to have a legal basis to process data, (iii) users’ right to download all their data, (iv) users’ right to require you to delete all their data. Other <a href="https://www.riskmanagementmonitor.com/canadas-own-gdpr-now-in-effect/">jurisdictions</a> are <a href="https://www.zdnet.com/article/australia-likely-to-get-its-own-gdpr/">exploring</a> similar rules.</li>
<li><strong>Data localization rules</strong>. <a href="https://economictimes.indiatimes.com/tech/internet/the-india-draft-bill-on-data-protection-draws-inspiration-from-gdpr-but-has-its-limits/articleshow/65173684.cms?from=mdr">India</a>, <a href="https://iapp.org/resources/topics/russias-data-localization-law/">Russia</a> and many other jurisdictions increasingly <a href="https://en.wikipedia.org/wiki/Data_localization">have or are exploring</a> rules that require data on users within the country to be stored inside the country. And even when explicit laws do not exist, there’s a growing shift toward concern (eg. <a href="https://qz.com/1613020/tiktok-might-be-a-chinese-cambridge-analytica-scale-privacy-threat/">1</a> <a href="https://thenextweb.com/podium/2019/03/09/eu-wants-tech-independence-from-the-us-but-itll-be-tricky/">2</a>) around data being moved to countries that are perceived to not sufficiently protect it.</li>
<li><strong>Sharing economy regulation</strong>. Sharing economy companies such as Uber <a href="https://www.theguardian.com/technology/2015/sep/11/uber-driver-employee-ruling">are having a hard time</a> arguing to courts that, given the extent to which their applications control and direct drivers’ activity, they should not be legally classified as employers.</li>
<li><strong>Cryptocurrency regulation</strong>. A <a href="https://www.systems.cs.cornell.edu/docs/fincen-cvc-guidance-final.pdf">recent FINCEN guidance</a> attempts to clarify what categories of cryptocurrency-related activity are and are not subject to regulatory licensing requirements in the United States. Running a hosted wallet? Regulated. Running a wallet where the user controls their funds? Not regulated. Running an anonymizing mixing service? If you’re <em>running</em> it, regulated. If you’re just writing code… <em>not regulated</em>.</li>
</ul>
<p>As <a href="https://twitter.com/el33th4xor/status/1126527690264195082">Emin Gun Sirer points out</a>, the FINCEN cryptocurrency guidance is not at all haphazard; rather, it’s trying to separate out categories of applications where the developer is actively controlling funds, from applications where the developer has no control. The guidance carefully separates out how <em>multisignature wallets</em>, where keys are held both by the operator and the user, are sometimes regulated and sometimes not:</p>
<blockquote>
<p>If the multiple-signature wallet provider restricts its role to creating un-hosted wallets that require adding a second authorization key to the wallet owner’s private key in order to validate and complete transactions, the provider is not a money transmitter because it does not accept and transmit value. On the other hand, if … the value is represented as an entry in the accounts of the provider, the owner does not interact with the payment system directly, or the provider maintains total independent control of the value, the provider will also qualify as a money transmitter.</p>
</blockquote>
<p>Although these events are taking place across a variety of contexts and industries, I would argue that there is a common trend at play. And the trend is this: <strong>control over users’ data and digital possessions and activity is rapidly moving from an asset to a liability</strong>. Before, every bit of control you have was good: it gives you more flexibility to earn revenue, if not now then in the future. Now, every bit of control you have is a liability: you might be regulated because of it. If you exhibit control over your users’ cryptocurrency, you are a money transmitter. If you have “sole discretion over fares, and can charge drivers a cancellation fee if they choose not to take a ride, prohibit drivers from picking up passengers not using the app and suspend or deactivate drivers’ accounts”, you are an employer. If you control your users’ data, you’re required to make sure you can argue just cause, have a compliance officer, and give your users access to download or delete the data.</p>
<p>If you are an application builder, and you are both lazy and fear legal trouble, there is one easy way to make sure that you violate none of the above new rules: <em>don’t build applications that centralize control</em>. If you build a wallet where the user holds their private keys, you really are still “just a software provider”. If you build a “decentralized Uber” that really is just a slick UI combining a payment system, a reputation system and a search engine, and don’t control the components yourself, you really won’t get hit by many of the same legal issues. If you build a website that just… doesn’t collect data (Static web pages? But that’s impossible!) you don’t have to even think about the GDPR.</p>
<p>This kind of approach is of course not realistic for everyone. There will continue to be many cases where going without the conveniences of centralized control simply sacrifices too much for both developers and users, and there are also cases where the business model considerations mandate a more centralized approach (eg. it’s easier to prevent non-paying users from using software if the software stays on your servers) win out. But we’re definitely very far from having explored the full range of possibilities that more decentralized approaches offer.</p>
<p>Generally, unintended consequences of laws, discouraging entire categories of activity when one wanted to only surgically forbid a few specific things, are considered to be a bad thing. Here though, I would argue that the forced shift in developers’ mindsets, from “I want to control more things just in case” to “I want to control fewer things just in case”, also has many positive consequences. Voluntarily giving up control, and voluntarily taking steps to deprive oneself of the ability to do mischief, does not come naturally to many people, and while ideologically-driven decentralization-maximizing projects exist today, it’s not at all obvious at first glance that such services will continue to dominate as the industry mainstreams. What this trend in regulation does, however, is that it gives a big nudge in favor of those applications that are willing to take the centralization-minimizing, user-sovereignty-maximizing “can’t be evil” route.</p>
<p>Hence, even though these regulatory changes are arguably not pro-freedom, at least if one is concerned with the freedom of application developers, and the transformation of the internet into a subject of political focus is bound to have many negative knock-on effects, the particular trend of control becoming a liability is in a strange way <em>even more pro-cypherpunk</em> (even if not intentionally!) than policies of maximizing total freedom for application developers would have been. Though the present-day regulatory landscape is very far from an optimal one from the point of view of almost anyone’s preferences, it has unintentionally dealt the movement for minimizing unneeded centralization and maximizing users’ control of their own assets, private keys and data a surprisingly strong hand to execute on its vision. And it would be highly beneficial to the movement to take advantage of it.</p>
Thu, 09 May 2019 18:03:10 -0700
https://vitalik.ca/general/2019/05/09/control_as_liability.html
https://vitalik.ca/general/2019/05/09/control_as_liability.htmlgeneralOn Free Speech<p><em>“A statement may be both true and dangerous. The previous sentence is such a statement.” - David Friedman</em></p>
<p>Freedom of speech is a topic that many internet communities have struggled with over the last two decades. Cryptocurrency and blockchain communities, a major part of their raison d’etre being censorship resistance, are especially poised to value free speech very highly, and yet, over the last few years, the extremely rapid growth of these communities and the very high financial and social stakes involved have repeatedly tested the application and the limits of the concept. In this post, I aim to disentangle some of the contradictions, and make a case what the norm of “free speech” really stands for.</p>
<h3 id="free-speech-laws-vs-free-speech">“Free speech laws” vs “free speech”</h3>
<p>A common, and in my own view frustrating, argument that I often hear is that “freedom of speech” is exclusively a legal restriction on what <em>governments</em> can act against, and has nothing to say regarding the actions of private entities such as corporations, privately-owned platforms, internet forums and conferences. One of the larger examples of “private censorship” in cryptocurrency communities was the decision of Theymos, the moderator of the <a href="http://reddit.com/r/bitcoin">/r/bitcoin</a> subreddit, to start heavily moderating the subreddit, forbidding arguments in favor of increasing the Bitcoin blockchain’s transaction capacity via a hard fork.</p>
<p><br /><center><img src="http://vitalik.ca/files/theymos.png" /></center><br /></p>
<p>Here is a timeline of the censorship as catalogued by John Blocke: <a href="https://medium.com/@johnblocke/a-brief-and-incomplete-history-of-censorship-in-r-bitcoin-c85a290fe43">https://medium.com/@johnblocke/a-brief-and-incomplete-history-of-censorship-in-r-bitcoin-c85a290fe43</a></p>
<p>Here is Theymos’s post defending his policies: <a href="https://www.reddit.com/r/Bitcoin/comments/3h9cq4/its_time_for_a_break_about_the_recent_mess">https://www.reddit.com/r/Bitcoin/comments/3h9cq4/its_time_for_a_break_about_the_recent_mess/</a>, including the now infamous line “If 90% of /r/Bitcoin users find these policies to be intolerable, then I want these 90% of /r/Bitcoin users to leave”.</p>
<p>A common strategy used by defenders of Theymos’s censorship was to say that heavy-handed moderation is okay because /r/bitcoin is “a private forum” owned by Theymos, and so he has the right to do whatever he wants in it; those who dislike it should move to other forums:</p>
<p><br /><center><img src="http://vitalik.ca/files/theymos2.png" /></center><br />
<br /><center><img src="http://vitalik.ca/files/theymos3.png" /></center><br /></p>
<p>And it’s true that Theymos has not <em>broken any laws</em> by moderating his forum in this way. But to most people, it’s clear that there is still some kind of free speech violation going on. So what gives? First of all, it’s crucially important to recognize that freedom of speech is not just a <em>law in some countries</em>. It’s also a social principle. And the underlying goal of the social principle is the same as the underlying goal of the law: to foster an environment where the ideas that win are ideas that are good, rather than just ideas that happen to be favored by people in a position of power. And governmental power is not the only kind of power that we need to protect from; there is also a corporation’s power to fire someone, an internet forum moderator’s power to <a href="https://cdn-images-1.medium.com/max/800/1*LPey4Z4mNwFE-ruiUkLYEw.png">delete almost every post in a discussion thread</a>, and many other kinds of power hard and soft.</p>
<p>So what is the underlying social principle here? <a href="https://www.lesswrong.com/posts/NCefvet6X3Sd4wrPc/uncritical-supercriticality">Quoting Eliezer Yudkowsky</a>:</p>
<blockquote>
<p>There are a very few injunctions in the human art of rationality that have no ifs, ands, buts, or escape clauses. This is one of them. Bad argument gets counterargument. Does not get bullet. Never. Never ever never for ever.</p>
</blockquote>
<p><a href="https://slatestarcodex.com/2013/12/29/the-spirit-of-the-first-amendment/">Slatestarcodex elaborates</a>:</p>
<blockquote>
<p>What does “bullet” mean in the quote above? Are other projectiles covered? Arrows? Boulders launched from catapults? What about melee weapons like swords or maces? Where exactly do we draw the line for “inappropriate responses to an argument”? A good response to an argument is one that addresses an idea; a bad argument is one that silences it. If you try to address an idea, your success depends on how good the idea is; if you try to silence it, your success depends on how powerful you are and how many pitchforks and torches you can provide on short notice. Shooting bullets is a good way to silence an idea without addressing it. So is firing stones from catapults, or slicing people open with swords, or gathering a pitchfork-wielding mob. But trying to get someone fired for holding an idea is also a way of silencing an idea without addressing it.</p>
</blockquote>
<p>That said, sometimes there is a rationale for “safe spaces” where people who, for whatever reason, just don’t want to deal with arguments of a particular type, can congregate and where those arguments actually do get silenced. Perhaps the most innocuous of all is spaces like <a href="http://ethresear.ch">ethresear.ch</a> where posts get silenced just for being “off topic” to keep the discussion focused. But there’s also a dark side to the concept of “safe spaces”; as <a href="https://www.popehat.com/2015/11/09/safe-spaces-as-shield-safe-spaces-as-sword/">Ken White writes</a>:</p>
<blockquote>
<p>This may come as a surprise, but I’m a supporter of ‘safe spaces.’ I support safe spaces because I support freedom of association. Safe spaces, if designed in a principled way, are just an application of that freedom… But not everyone imagines “safe spaces” like that. Some use the concept of “safe spaces” as a sword, wielded to annex public spaces and demand that people within those spaces conform to their private norms. That’s not freedom of association</p>
</blockquote>
<p>Aha. So making your own safe space off in a corner is totally fine, but there is also this concept of a “public space”, and trying to turn a public space into a safe space for one particular special interest is wrong. So what is a “public space”? It’s definitely clear that a public space is <em>not</em> just “a space owned and/or run by a government”; the concept of <a href="https://en.wikipedia.org/wiki/Privately_owned_public_space">privately owned public spaces</a> is a well-established one. This is true even informally: it’s a common moral intuition, for example, that it’s less bad for a private individual to commit violations such as discriminating against races and genders than it is for, say, a shopping mall to do the same. In the case or the /r/bitcoin subreddit, one can make the case, regardless of who technically owns the top moderator position in the subreddit, that the subreddit very much is a public space. A few arguments particularly stand out:</p>
<ul>
<li>It occupies “prime real estate”, specifically the word “bitcoin”, which makes people consider it to be <em>the</em> default place to discuss Bitcoin.</li>
<li>The value of the space was created not just by Theymos, but by thousands of people who arrived on the subreddit to discuss Bitcoin with an implicit expectation that it is, and will continue, to be a public space for discussing Bitcoin.</li>
<li>Theymos’s shift in policy was a surprise to many people, and it was <em>not</em> foreseeable ahead of time that it would take place.</li>
</ul>
<p>If, instead, Theymos had created a subreddit called /r/bitcoinsmallblockers, and explicitly said that it was a curated space for small block proponents and attempting to instigate controversial hard forks was not welcome, then it seems likely that very few people would have seen anything wrong about this. They would have opposed his ideology, but few (at least in blockchain communities) would try to claim that it’s <em>improper</em> for people with ideologies opposed to their own to have spaces for internal discussion. But back in reality, Theymos tried to “annex a public space and demand that people within the space confirm to his private norms”, and so we have the Bitcoin community block size schism, a highly acrimonious fork and chain split, and now a cold peace between Bitcoin and Bitcoin Cash.</p>
<h3 id="deplatforming">Deplatforming</h3>
<p>About a year ago at Deconomy I publicly shouted down Craig Wright, <a href="https://github.com/vbuterin/cult-of-craig">a scammer claiming to be Satoshi Nakamoto</a>, finishing my explanation of why the things he says make no sense with the question “why is this fraud allowed to speak at this conference?”</p>
<p><br /><center><a href="https://www.youtube.com/watch?v=WaWcJPSs9Yw&feature=youtu.be&t=20m33s"><img src="http://vitalik.ca/files/me_against_craig.png" style="width:600px" /></a></center><br /></p>
<p>Of course, Craig Wright’s partisans replied back with…. <a href="https://coingeek.com/samson-mow-vitalik-buterin-exposed/">accusations of censorship</a>:</p>
<p><br /><center><img src="http://vitalik.ca/files/craigwright.png" /></center><br /></p>
<p>Did I try to “silence” Craig Wright? I would argue, no. One could argue that this is because “Deconomy is not a public space”, but I think the much better argument is that a conference is fundamentally different from an internet forum. An internet forum can actually try to be a fully neutral medium for discussion where anything goes; a conference, on the other hand, is by its very nature a highly curated list of presentations, allocating a limited number of speaking slots and actively channeling a large amount of attention to those lucky enough to get a chance to speak. A conference is an editorial act by the organizers, saying “here are some ideas and views that we think people really should be exposed to and hear”. Every conference “censors” almost every viewpoint because there’s not enough space to give them all a chance to speak, and this is inherent to the format; so raising an objection to a conference’s judgement in making its selections is absolutely a legitimate act.</p>
<p>This extends to other kinds of selective platforms. Online platforms such as Facebook, Twitter and Youtube already engage in active selection through algorithms that influence what people are more likely to be recommended. Typically, they do this for selfish reasons, setting up their algorithms to maximize “engagement” with their platform, often with unintended byproducts like <a href="https://www.independent.co.uk/life-style/gadgets-and-tech/flat-earth-youtube-conspiracy-theory-videos-research-study-a8783091.html">promoting flat earth conspiracy theories</a>. So given that these platforms are already engaging in (automated) selective presentation, it seems eminently reasonable to criticize them for not directing these same levers toward more pro-social objectives, or at the least pro-social objectives that all major reasonable political tribes agree on (eg. quality intellectual discourse). Additionally, the “censorship” doesn’t seriously block anyone’s ability to learn Craig Wright’s side of the story; you can just go visit their website, here you go: <a href="https://coingeek.com/">https://coingeek.com/</a>. <strong>If someone is already operating a platform that makes editorial decisions, asking them to make such decisions with the same magnitude but with more pro-social criteria seems like a very reasonable thing to do</strong>.</p>
<p>A more recent example of this principle at work is the #DelistBSV campaign, where some cryptocurrency exchanges, most famously <a href="https://support.binance.com/hc/en-us/articles/360026666152">Binance</a>, removed support for trading BSV (the Bitcoin fork promoted by Craig Weight). Once again, many people, even <a href="https://decryptmedia.com/6552/binance-kraken-delisting-bitcoin-sv-sets-bad-precedent">reasonable people</a>, accused this campaign of being an <a href="https://twitter.com/angela_walch/status/1117921461304475649">exercise in censorship</a>, raising parallels to credit card companies blocking Wikileaks:</p>
<p><br /><center><img src="http://vitalik.ca/files/craigwright2.png" /></center><br /></p>
<p>I personally have been a <a href="https://techcrunch.com/2018/07/06/vitalik-buterin-i-definitely-hope-centralized-exchanges-go-burn-in-hell-as-much-as-possible/">critic of the power wielded by centralized exchanges</a>. Should I oppose #DelistBSV on free speech grounds? I would argue no, it’s ok to support it, but this is definitely a much closer call.</p>
<p>Many #DelistBSV participants like Kraken are definitely not “anything-goes” platforms; they already make many editorial decisions about which currencies they accept and refuse. Kraken only <a href="https://trade.kraken.com/markets">accepts about a dozen currencies</a>, so they are passively “censoring” almost everyone. Shapeshift supports more currencies but it does not support <a href="https://spankchain.com/">SPANK</a>, or even <a href="https://kyber.network/">KNC</a>. So in these two cases, delisting BSV is more like reallocation of a scarce resource (attention/legitimacy) than it is censorship. Binance is a bit different; it does accept a very large array of cryptocurrencies, adopting a philosophy much closer to anything-goes, and it does have a unique position as market leader with a lot of liquidity.</p>
<p>That said, one can argue two things in Binance’s favor. First of all, censorship is retaliating against a truly malicious exercise of censorship on the part of core BSV community members when they threatened critics like Peter McCormack with legal letters (see <a href="https://twitter.com/PeterMcCormack/status/1117448742892986368">Peter’s response</a>); in “anarchic” environments with large disagreements on what the norms are, “an eye for an eye” in-kind retaliation is one of the better social norms to have because it ensures that people only face punishments that they in some sense have through their own actions demonstrated they believe are legitimate. Furthermore, the delistings won’t make it that hard for people to buy or sell BSV; Coinex has said that <a href="https://twitter.com/yhaiyang/status/1118002345961353216">they will not delist</a> (and I would actually oppose second-tier “anything-goes” exchanges delisting). But the delistings <em>do</em> send a strong message of social condemnation of BSV, which is useful and needed. So there’s a case to support all delistings so far, though on reflection Binance refusing to delist “because freedom” would have also been not as unreasonable as it seems at first glance.</p>
<p>It’s in general absolutely potentially reasonable to oppose the existence of a concentration of power, but support that concentration of power being used for purposes that you consider prosocial as long as that concentration exists; see Bryan Caplan’s exposition on <a href="https://www.econlib.org/archives/2014/10/ebola_and_open.html">reconciling</a> supporting open borders and also supporting anti-ebola restrictions for an example in a different field. Opposing concentrations of power only requires that one believe those concentrations of power to be <em>on balance</em> harmful and abusive; it does not mean that one must oppose <em>all</em> things that those concentrations of power do.</p>
<p>If someone manages to make a <em>completely permissionless</em> cross-chain decentralized exchange that facilitates trade between any asset and any other asset, then being “listed” on the exchange would <em>not</em> send a social signal, because everyone is listed; and I would support such an exchange existing even if it supports trading BSV. The thing that I do support is BSV being removed from already exclusive positions that confer higher tiers of legitimacy than simple existence.</p>
<p>So to conclude: censorship in public spaces bad, even if the public spaces are non-governmental; censorship in genuinely private spaces (especially spaces that are <em>not</em> “defaults” for a broader community) can be okay; ostracizing projects with the goal and effect of denying access to them, bad; ostracizing projects with the goal and effect of denying them scarce legitimacy can be okay.</p>
Tue, 16 Apr 2019 18:03:10 -0700
https://vitalik.ca/general/2019/04/16/free_speech.html
https://vitalik.ca/general/2019/04/16/free_speech.htmlgeneralOn Collusion<p><em>Special thanks to Glen Weyl, Phil Daian and Jinglan Wang for review</em></p>
<p>Over the last few years there has been an increasing interest in using deliberately engineered economic incentives and mechanism design to align behavior of participants in various contexts. In the blockchain space, mechanism design first and foremost provides the security for the blockchain itself, encouraging miners or proof of stake validators to participate honestly, but more recently it is being applied in <a href="https://www.augur.net/">prediction markets</a>, “<a href="https://medium.com/@tokencuratedregistry/a-simple-overview-of-token-curated-registries-84e2b7b19a06">token curated registries</a>” and many other contexts. The nascent <a href="https://radicalxchange.org/">RadicalXChange movement</a> has meanwhile spawned experimentation with <a href="https://medium.com/@simondlr/this-artwork-is-always-on-sale-92a7d0c67f43">Harberger taxes</a>, quadratic voting, <a href="https://medium.com/gitcoin/gitcoin-grants-50k-open-source-fund-e20e09dc2110">quadratic financing</a> and more. More recently, there has also been growing interest in using token-based incentives to try to encourage quality posts in social media. However, as development of these systems moves closer from theory to practice, there are a number of challenges that need to be addressed, challenges that I would argue have not yet been adequately confronted.</p>
<p>As a recent example of this move from theory toward deployment, Bihu, a Chinese platform that has recently released a coin-based mechanism for encouraging people to write posts. The basic mechanism (see whitepaper in Chinese <a href="https://www.chainwhy.com/whitepaper/keywhitepaper.html">here</a>) is that if a user of the platform holds KEY tokens, they have the ability to stake those KEY tokens on articles; every user can make <code class="highlighter-rouge">k</code> “upvotes” per day, and the “weight” of each upvote is proportional to the stake of the user making the upvote. Articles with a greater quantity of stake upvoting them appear more prominently, and the author of an article gets a reward of KEY tokens roughly proportional to the quantity of KEY upvoting that article. This is an oversimplification and the actual mechanism has some nonlinearities baked into it, but they are not essential to the basic functioning of the mechanism. KEY has value because it can be used in various ways inside the platform, but particularly a percentage of all ad revenues get used to buy and burn KEY (yay, big thumbs up to them for doing this and not making yet another <a href="https://vitalik.ca/general/2017/10/17/moe.html">medium of exchange token</a>!).</p>
<p>This kind of design is far from unique; incentivizing online content creation is something that very many people care about, and there have been many designs of a similar character, as well as some fairly different designs. And in this case this particular platform is already being used significantly:</p>
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<img src="https://vitalik.ca/files/screenie.png" />
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<p>A few months ago, the Ethereum trading subreddit <a href="http://reddit.com/r/ethtrader">/r/ethtrader</a> introduced a somewhat similar experimental feature where a token called “donuts” is issued to users that make comments that get upvoted, with a set amount of donuts issued weekly to users in proportion to how many upvotes their comments received. The donuts could be used to buy the right to set the contents of the banner at the top of the subreddit, and could also be used to vote in community polls. However, unlike what happens in the KEY system, here the reward that B receives when A upvotes B is not proportional to A’s existing coin supply; instead, each Reddit account has an equal ability to contribute to other Reddit accounts.</p>
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<img src="https://vitalik.ca/files/donuts.png" />
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<p><br /></p>
<p>These kinds of experiments, attempting to reward quality content creation in a way that goes beyond the known limitations of donations/microtipping, are very valuable; under-compensation of user-generated internet content is a very significant problem in society in general (see “<a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3243656">liberal radicalism</a>” and “<a href="http://radicalmarkets.com/chapters/data-as-labor/">data as labor</a>”), and it’s heartening to see crypto communities attempting to use the power of mechanism design to make inroads on solving it. <strong>But unfortunately, these systems are also vulnerable to attack.</strong></p>
<h3 id="self-voting-plutocracy-and-bribes">Self-voting, plutocracy and bribes</h3>
<p>Here is how one might economically attack the design proposed above. Suppose that some wealthy user acquires some quantity <code class="highlighter-rouge">N</code> of tokens, and as a result each of the user’s <code class="highlighter-rouge">k</code> upvotes gives the recipient a reward of <code class="highlighter-rouge">N * q</code> (<code class="highlighter-rouge">q</code> here probably being a very small number, eg. think <code class="highlighter-rouge">q = 0.000001</code>). The user simply upvotes their own sockpuppet accounts, giving themselves the reward of <code class="highlighter-rouge">N * k * q</code>. Then, the system simply collapses into each user having an “interest rate” of <code class="highlighter-rouge">k * q</code> per period, and the mechanism accomplishes nothing else.</p>
<p>The actual Bihu mechanism seemed to anticipate this, and has some superlinear logic where articles with more KEY upvoting them gain a disproportionately greater reward, seemingly to encourage upvoting popular posts rather than self-upvoting. It’s a common pattern among coin voting governance systems to add this kind of superlinearity to prevent self-voting from undermining the entire system; most DPOS schemes have a limited number of delegate slots with zero rewards for anyone who does not get enough votes to join one of the slots, with similar effect. But these schemes invariably introduce two new weaknesses:</p>
<ul>
<li>They <strong>subsidize plutocracy</strong>, as very wealthy individuals and cartels can still get enough funds to self-upvote.</li>
<li>They can be circumvented by users <strong><em>bribing</em></strong> other users to vote for them en masse.</li>
</ul>
<p>Bribing attacks may sound farfetched (who here has ever accepted a bribe in real life?), but in a mature ecosystem they are much more realistic than they seem. In most <a href="https://vitalik.ca/general/2017/12/17/voting.html">contexts where bribing has taken place</a> in the blockchain space, the operators use a euphemistic new name to give the concept a friendly face: it’s not a bribe, it’s a “staking pool” that “shares dividends”. Bribes can even be obfuscated: imagine a cryptocurrency exchange that offers zero fees and spends the effort to make an abnormally good user interface, and does not even try to collect a profit; instead, it uses coins that users deposit to participate in various coin voting systems. There will also inevitably be people that see in-group collusion as just plain normal; see a recent <a href="https://twitter.com/MapleLeafCap/status/1044958643731533825">scandal involving EOS DPOS</a> for one example:</p>
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<a href="https://twitter.com/MapleLeafCap/status/1044958647535767552"><img src="http://vitalik.ca/files/mapleleaf1.png" style="width:480px" /></a>
<a href="https://twitter.com/MapleLeafCap/status/1044958649188327429"><img src="http://vitalik.ca/files/mapleleaf2.png" style="width:480px" /></a>
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<p><br /></p>
<p>Finally, there is the possibility of a “negative bribe”, ie. blackmail or coercion, threatening participants with harm unless they act inside the mechanism in a certain way.</p>
<p>In the /r/ethtrader experiment, fear of people coming in and <em>buying</em> donuts to shift governance polls led to the community deciding to make only locked (ie. untradeable) donuts eligible for use in voting. But there’s an even cheaper attack than buying donuts (an attack that can be thought of as a kind of obfuscated bribe): <em>renting</em> them. If an attacker is already holding ETH, they can use it as collateral on a platform like <a href="https://compound.finance/">Compound</a> to take out a loan of some token, giving you the full right to use that token for whatever purpose including participating in votes, and when they’re done they simply send the tokens back to the loan contract to get their collateral back - all without having to endure even a second of price exposure to the token that they just used to swing a coin vote, even if the coin vote mechanism includes a time lockup (as eg. Bihu does). In every case, issues around bribing, and accidentally over-empowering well-connected and wealthy participants, prove surprisingly difficult to avoid.</p>
<h3 id="identity">Identity</h3>
<p>Some systems attempt to mitigate the plutocratic aspects of coin voting by making use of an identity system. In the case of the /r/ethtrader donut system, for example, although <em>governance polls</em> are done via coin vote, the mechanism that determines <em>how many donuts (ie. coins) you get in the first place</em> is based on Reddit accounts: 1 upvote from 1 Reddit account = N donuts earned. The ideal goal of an identity system is to make it relatively easy for individuals to get one identity, but relatively difficult to get many identities. In the /r/ethtrader donut system, that’s Reddit accounts, in the Gitcoin CLR matching gadget, it’s Github accounts that are used for the same purpose. But identity, at least the way it has been implemented so far, is a fragile thing….</p>
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<a href="https://twitter.com/JamieJBartlett/status/1105151495773847552"><img src="http://vitalik.ca/files/clickfarm.png" style="width:400px" /></a>
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<p>Oh, are you too lazy to make a big rack of phones? Well maybe you’re looking <a href="http://buyaccs.com">for this</a>:</p>
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<a href="http://buyaccs.com"><img src="http://vitalik.ca/files/buyaccs.png" style="width:500px" /></a><br /><br />
<small><i>Usual warning about how sketchy sites may or may not scam you, do your own research, etc. etc. applies.</i></small>
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<p><br /></p>
<p>Arguably, attacking these mechanisms by simply controlling thousands of fake identities like a puppetmaster is <em>even easier</em> than having to go through the trouble of bribing people. And if you think the response is to just increase security to go up to <em>government-level</em> IDs? Well, if you want to get a few of those you can start exploring <a href="https://thehiddenwiki.com/Main_Page">here</a>, but keep in mind that there are specialized criminal organizations that are well ahead of you, and even if all the underground ones are taken down, hostile governments are definitely going to create fake passports by the millions if we’re stupid enough to create systems that make that sort of activity profitable. And this doesn’t even begin to mention attacks in the opposite direction, identity-issuing institutions attempting to disempower marginalized communities by <em>denying</em> them identity documents…</p>
<h4 id="collusion">Collusion</h4>
<p>Given that so many mechanisms seem to fail in such similar ways once multiple identities or even liquid markets get into the picture, one might ask, is there some deep common strand that causes all of these issues? I would argue the answer is yes, and the “common strand” is this: it is much harder, and more likely to be outright impossible, to make mechanisms that maintain desirable properties in a model where participants can collude, than in a model where they can’t. Most people likely already have some intuition about this; specific instances of this principle are behind well-established norms and often laws promoting competitive markets and restricting price-fixing cartels, vote buying and selling, and bribery. But the issue is much deeper and more general.</p>
<p>In the version of game theory that focuses on individual choice - that is, the version that assumes that each participant makes decisions independently and that does not allow for the possibility of groups of agents working as one for their mutual benefit, there are <a href="https://en.wikipedia.org/wiki/Nash_equilibrium#Proof_of_existence">mathematical proofs</a> that at least one stable Nash equilibrium must exist in any game, and mechanism designers have a very wide latitude to “engineer” games to achieve specific outcomes. But in the version of game theory that allows for the possibility of coalitions working together, called <em>cooperative game theory</em>, <strong>there are <a href="https://en.wikipedia.org/wiki/Bondareva%E2%80%93Shapley_theorem">large classes of games</a> that do not have any stable outcome that a coalition cannot profitably deviate from</strong>.</p>
<p><em>Majority games</em>, formally described as games of <code class="highlighter-rouge">N</code> agents where any subset of more than half of them can capture a fixed reward and split it among themselves, a setup eerily similar to many situations in corporate governance, politics and many other situations in human life, are <a href="https://web.archive.org/web/20180329012328/https://www.math.mcgill.ca/vetta/CS764.dir/Core.pdf">part of that set of inherently unstable games</a>. That is to say, if there is a situation with some fixed pool of resources and some currently established mechanism for distributing those resources, and it’s unavoidably possible for 51% of the participants can conspire to seize control of the resources, no matter what the current configuration is there is always some conspiracy that can emerge that would be profitable for the participants. However, that conspiracy would then in turn be vulnerable to potential new conspiracies, possibly including a combination of previous conspirators and victims… and so on and so forth.</p>
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<table>
<tr><td>Round</td><td>A</td><td>B</td><td>C</td></tr>
<tr><td>1</td><td>1/3</td><td>1/3</td><td>1/3</td></tr>
<tr><td>2</td><td style="background-color:grey">1/2</td><td style="background-color:grey">1/2</td><td>0</td></tr>
<tr><td>3</td><td style="background-color:grey">2/3</td><td>0</td><td style="background-color:grey">1/3</td></tr>
<tr><td>4</td><td>0</td><td style="background-color:grey">1/3</td><td style="background-color:grey">2/3</td></tr>
</table>
</center>
<p><br /></p>
<p><strong>This fact, the instability of majority games under cooperative game theory, is arguably highly underrated as a simplified general mathematical model of why there may well be no “end of history” in politics and no system that proves fully satisfactory; I personally believe it’s much more useful than the more famous <a href="https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem">Arrow’s theorem</a>, for example.</strong></p>
<p>There are two ways to get around this issue. The first is to try to restrict ourselves to the class of games that <em>are</em> “identity-free” and “collusion-safe”, so where we do not need to worry about either bribes or identities. The second is to try to attack the identity and collusion resistance problems directly, and actually solve them well enough that we can implement non-collusion-safe games with the richer properties that they offer.</p>
<h3 id="identity-free-and-collusion-safe-game-design">Identity-free and collusion-safe game design</h3>
<p>The class of games that is identity-free and collusion-safe is substantial. Even proof of work is collusion-safe up to the bound of a single actor having <a href="https://arxiv.org/abs/1507.06183">~23.21% of total hashpower</a>, and this bound can be increased up to 50% with <a href="https://eprint.iacr.org/2016/916.pdf">clever engineering</a>. Competitive markets are reasonably collusion-safe up until a relatively high bound, which is easily reached in some cases but in other cases is not.</p>
<p>In the case of <em>governance</em> and <em>content curation</em> (both of which are really just special cases of the general problem of identifying public goods and public bads) a major class of mechanism that works well is <em><a href="https://blog.ethereum.org/2014/08/21/introduction-futarchy/">futarchy</a></em> - typically portrayed as “governance by prediction market”, though I would also argue that the use of security deposits is fundamentally in the same class of technique. The way futarchy mechanisms, in their most general form, work is that they make “voting” not just an expression of opinion, but also a <em>prediction</em>, with a reward for making predictions that are true and a penalty for making predictions that are false. For example, <a href="https://ethresear.ch/t/prediction-markets-for-content-curation-daos/1312">my proposal</a> for “prediction markets for content curation DAOs” suggests a semi-centralized design where anyone can upvote or downvote submitted content, with content that is upvoted more being more visible, where there is also a “moderation panel” that makes final decisions. For each post, there is a small probability (proportional to the total volume of upvotes+downvotes on that post) that the moderation panel will be called on to make a final decision on the post. If the moderation panel approves a post, everyone who upvoted it is rewarded and everyone who downvoted it is penalized, and if the moderation panel disapproves a post the reverse happens; this mechanism encourages participants to make upvotes and downvotes that try to “predict” the moderation panel’s judgements.</p>
<p>Another possible example of futarchy is a governance system for a project with a token, where anyone who votes for a decision is obligated to purchase some quantity of tokens at the price at the time the vote begins if the vote wins; this ensures that voting on a bad decision is costly, and in the limit if a bad decision wins a vote everyone who approved the decision must essentially buy out everyone else in the project. This ensures that an individual vote for a “wrong” decision can be very costly for the voter, precluding the possibility of cheap bribe attacks.</p>
<p><br /></p>
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<img src="https://ethresear.ch/uploads/default/original/2X/4/4236db5226633dcc00bb4924f55db33488707488.png" style="width:600px" /><br />
<small><i>A graphical description of one form of futarchy, creating two markets representing the two "possible future worlds" and picking the one with a more favorable price. Source <a href="https://ethresear.ch/uploads/default/original/2X/4/4236db5226633dcc00bb4924f55db33488707488.png">this post on ethresear.ch</a></i></small>
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<p><br /></p>
<p>However, that range of things that mechanisms of this type can do is limited. In the case of the content curation example above, we’re not really solving governance, we’re just <em>scaling</em> the functionality of a governance gadget that is already assumed to be trusted. One could try to replace the moderation panel with a prediction market on the price of a token representing the right to purchase advertising space, but in practice prices are too noisy an indicator to make this viable for anything but a very small number of very large decisions. And often the value that we’re trying to maximize is explicitly something other than maximum value of a coin.</p>
<p>Let’s take a more explicit look at why, in the more general case where we can’t easily determine the value of a governance decision via its impact on the price of a token, good mechanisms for identifying public goods and bads unfortunately cannot be identity-free or collusion-safe. If one tries to preserve the property of a game being identity-free, building a system where identities don’t matter and only coins do, <strong>there is an impossible tradeoff between either failing to incentivize legitimate public goods or over-subsidizing plutocracy</strong>.</p>
<p>The argument is as follows. Suppose that there is some author that is producing a public good (eg. a series of blog posts) that provides value to each member of a community of 10000 people. Suppose there exists some mechanism where members of the community can take an action that causes the author to receive a gain of $1. Unless the community members are <em>extremely</em> altruistic, for the mechanism to work the cost of taking this action must be much lower than $1, as otherwise the portion of the benefit captured by the member of the community supporting the author would be much smaller than the cost of supporting the author, and so the system collapses into a <a href="https://en.wikipedia.org/wiki/Tragedy_of_the_commons">tragedy of the commons</a> where no one supports the author. Hence, there must exist a way to cause the author to earn $1 at a cost much less than $1. But now suppose that there is also a fake community, which consists of 10000 fake sockpuppet accounts of the same wealthy attacker. This community takes all of the same actions as the real community, except instead of supporting the author, they support <em>another</em> fake account which is also a sockpuppet of the attacker. If it was possible for a member of the “real community” to give the author $1 at a personal cost of much less than $1, it’s possible for the attacker to give <em>themselves</em> $1 at a cost much less than $1 over and over again, and thereby drain the system’s funding. Any mechanism that can help genuinely under-coordinated parties coordinate will, without the right safeguards, also help already coordinated parties (such as many accounts controlled by the same person) <em>over-coordinate</em>, extracting money from the system.</p>
<p>A similar challenge arises when the goal is not funding, but rather determining what content should be most visible. What content do you think would get more dollar value supporting it: a legitimately high quality blog article benefiting thousands of people but benefiting each individual person relatively slightly, or this?</p>
<p><br /></p>
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<img src="https://vitalik.ca/files/cocacola.jpg" style="width:550px" />
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<p><br /></p>
<p>Or perhaps this?</p>
<p><br /></p>
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<img src="https://vitalik.ca/files/bitconnect.png" style="width:550px" />
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<p><br /></p>
<p>Those who have been following recent politics “in the real world” might also point out a different kind of content that benefits highly centralized actors: social media manipulation by hostile governments. Ultimately, both centralized systems and decentralized systems are facing the same fundamental problem, which is that <strong>the “marketplace of ideas” (and of public goods more generally) is very far from an “efficient market” in the sense that economists normally use the term</strong>, and this leads to both underproduction of public goods even in “peacetime” but also vulnerability to active attacks. It’s just a hard problem.</p>
<p>This is also why coin-based voting systems (like Bihu’s) have one major genuine advantage over identity-based systems (like the Gitcoin CLR or the /r/ethtrader donut experiment): at least there is no benefit to buying accounts en masse, because everything you do is proportional to how many coins you have, regardless of how many accounts the coins are split between. However, mechanisms that do not rely on any model of identity and only rely on coins fundamentally cannot solve the problem of concentrated interests outcompeting dispersed communities trying to support public goods; an identity-free mechanism that empowers distributed communities cannot avoid over-empowering centralized plutocrats pretending to be distributed communities.</p>
<p>But it’s not just identity issues that public goods games are vulnerable too; it’s also bribes. To see why, consider again the example above, but where instead of the “fake community” being 10001 sockpuppets of the attacker, the attacker only has one identity, the account receiving funding, and the other 10000 accounts are real users - but users that receive a bribe of $0.01 each to take the action that would cause the attacker to gain an additional $1. As mentioned above, these bribes can be highly obfuscated, even through third-party custodial services that vote on a user’s behalf in exchange for convenience, and in the case of “coin vote” designs an obfuscated bribe is even easier: one can do it by renting coins on the market and using them to participate in votes. Hence, while some kinds of games, particularly prediction market or security deposit based games, can be made collusion-safe and identity-free, generalized public goods funding seems to be a class of problem where collusion-safe and identity-free approaches unfortunately just cannot be made to work.</p>
<h3 id="collusion-resistance-and-identity">Collusion resistance and identity</h3>
<p>The other alternative is attacking the identity problem head-on. As mentioned above, simply going up to higher-security centralized identity systems, like passports and other government IDs, will not work at scale; in a sufficiently incentivized context, they are very insecure and vulnerable to the issuing governments themselves! Rather, the kind of “identity” we are talking about here is some kind of robust multifactorial set of claims that an actor identified by some set of messages actually is a unique individual. A very early proto-model of this kind of networked identity is arguably social recovery in HTC’s blockchain phone:</p>
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<img src="https://vitalik.ca/files/htcphone.jpg" style="width:300px" />
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<p><br /></p>
<p>The basic idea is that your private key is secret-shared between up to five trusted contacts, in such a way that mathematically ensures that three of them can recover the original key, but two or fewer can’t. This qualifies as an “identity system” - it’s your five friends determining whether or not someone trying to recover your account actually is you. However, it’s a special-purpose identity system trying to solve a problem - personal account security - that is different from (and easier than!) the problem of attempting to identify unique humans. That said, the general model of individuals making claims about each other can quite possibly be bootstrapped into some kind of more robust identity model. These systems could be augmented if desired using the “futarchy” mechanic described above: if someone makes a claim that someone is a unique human, and someone else disagrees, and both sides are willing to put down a bond to litigate the issue, the system can call together a judgement panel to determine who is right.</p>
<p>But we also want another crucially important property: we want an identity that you cannot credibly rent or sell. Obviously, we can’t prevent people from making a deal “you send me $50, I’ll send you my key”, but what we <em>can</em> try to do is prevent such deals from being <em>credible</em> - make it so that the seller can easily cheat the buyer and give the buyer a key that doesn’t actually work. One way to do this is to make a mechanism by which the owner of a key can send a transaction that revokes the key and replaces it with another key of the owner’s choice, all in a way that cannot be proven. Perhaps the simplest way to get around this is to either use a trusted party that runs the computation and only publishes results (along with zero knowledge proofs proving the results, so the trusted party is trusted only for privacy, not integrity), or decentralize the same functionality through <a href="https://blog.ethereum.org/2014/12/26/secret-sharing-daos-crypto-2-0/">multi-party computation</a>. Such approaches will not solve collusion completely; a group of friends could still come together and sit on the same couch and coordinate votes, but they will at least reduce it to a manageable extent that will not lead to these systems outright failing.</p>
<p>There is a further problem: initial distribution of the key. What happens if a user creates their identity inside a third-party custodial service that then stores the private key and uses it to clandestinely make votes on things? This would be an implicit bribe, the user’s voting power in exchange for providing to the user a convenient service, and what’s more, if the system is secure in that it successfully prevents bribes by making votes unprovable, clandestine voting by third-party hosts would <em>also</em> be undetectable. The only approach that gets around this problem seems to be…. in-person verification. For example, one could have an ecosystem of “issuers” where each issuer issues smart cards with private keys, which the user can immediately download onto their smartphone and send a message to replace the key with a different key that they do not reveal to anyone. These issuers could be meetups and conferences, or potentially individuals that have already been deemed by some voting mechanic to be trustworthy.</p>
<p>Building out the infrastructure for making collusion-resistant mechanisms possible, including robust decentralized identity systems, is a difficult challenge, but if we want to unlock the potential of such mechanisms, it seems unavoidable that we have to do our best to try. It is true that the current computer-security dogma around, for example, introducing online voting is simply “<a href="https://www.geekwire.com/2018/online-voting-dont-experts-say-report-americas-election-system-security/">don’t</a>”, but if we want to expand the role of voting-like mechanisms, including more advanced forms such as quadratic voting and quadratic finance, to more roles, we have no choice but to confront the challenge head-on, try really hard, and hopefully succeed at making something secure enough, for at least some use cases.</p>
Wed, 03 Apr 2019 18:03:10 -0700
https://vitalik.ca/general/2019/04/03/collusion.html
https://vitalik.ca/general/2019/04/03/collusion.htmlgeneralA CBC Casper Tutorial<p><em>Special thanks to Vlad Zamfir, Aditya Asgaonkar, Ameen Soleimani and Jinglan Wang for review</em></p>
<p>In order to help more people understand “the other Casper” (Vlad Zamfir’s CBC Casper), and specifically the instantiation that works best for blockchain protocols, I thought that I would write an explainer on it myself, from a less abstract and more “close to concrete usage” point of view. Vlad’s descriptions of CBC Casper can be found <a href="https://www.youtube.com/watch?v=GNGbd_RbrzE">here</a> and <a href="https://github.com/ethereum/cbc-casper/wiki/FAQ">here</a> and <a href="https://github.com/cbc-casper/cbc-casper-paper">here</a>; you are welcome and encouraged to look through these materials as well.</p>
<p>CBC Casper is designed to be fundamentally very versatile and abstract, and come to consensus on pretty much any data structure; you can use CBC to decide whether to choose 0 or 1, you can make a simple block-by-block chain run on top of CBC, or a 2<sup>92</sup>-dimensional hypercube tangle DAG, and pretty much anything in between.</p>
<p>But for simplicity, we will first focus our attention on one concrete case: a simple chain-based structure. We will suppose that there is a fixed validator set consisting of N validators (a fancy word for “staking nodes”; we also assume that each node is staking the same amount of coins, cases where this is not true can be simulated by assigning some nodes multiple validator IDs), time is broken up into ten-second slots, and validator <code class="highlighter-rouge">k</code> can create a block in slot <code class="highlighter-rouge">k</code>, <code class="highlighter-rouge">N + k</code>, <code class="highlighter-rouge">2N + k</code>, etc. Each block points to one specific parent block. Clearly, if we wanted to make something maximally simple, we could just take this structure, impose a longest chain rule on top of it, and call it a day.</p>
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<img src="https://vitalik.ca/files/Chain3.png" /><br />
<small><i>The green chain is the longest chain (length 6) so it is considered to be the "canonical chain".</i></small>
</center>
<p><br /></p>
<p>However, what we care about here is adding some notion of “finality” - the idea that some block can be so firmly established in the chain that it cannot be overtaken by a competing block unless a very large portion (eg. 1/4) of validators commit a <em>uniquely attributable fault</em> - act in some way which is clearly and cryptographically verifiably malicious. If a very large portion of validators <em>do</em> act maliciously to revert the block, proof of the misbehavior can be submitted to the chain to take away those validators’ entire deposits, making the reversion of finality extremely expensive (think hundreds of millions of dollars).</p>
<h3 id="lmd-ghost">LMD GHOST</h3>
<p>We will take this one step at a time. First, we replace the fork choice rule (the rule that chooses which chain among many possible choices is “the canonical chain”, ie. the chain that users should care about), moving away from the simple longest-chain-rule and instead using “latest message driven GHOST”. To show how LMD GHOST works, we will modify the above example. To make it more concrete, suppose the validator set has size 5, which we label A, B, C, D, E, so validator A makes the blocks at slots 0 and 5, validator B at slots 1 and 6, etc. A client evaluating the LMD GHOST fork choice rule cares only about the most recent (ie. highest-slot) message (ie. block) signed by each validator:</p>
<center>
<img src="https://vitalik.ca/files/Chain4.png" /><br />
<small><i>Latest messages in blue, slots from left to right (eg. A's block on the left is at slot 0, etc.)</i></small>
</center>
<p><br /></p>
<p>Now, we will use only these messages as source data for the “greedy heaviest observed subtree” (GHOST) fork choice rule: start at the genesis block, then each time there is a fork choose the side where more of the latest messages support that block’s subtree (ie. more of the latest messages support either that block or one of its descendants), and keep doing this until you reach a block with no children. We can compute for each block the subset of latest messages that support either the block or one of its descendants:</p>
<center>
<img src="https://vitalik.ca/files/Chain5.png" /><br />
</center>
<p>Now, to compute the head, we start at the beginning, and then at each fork pick the higher number: first, pick the bottom chain as it has 4 latest messages supporting it versus 1 for the single-block top chain, then at the next fork support the middle chain. The result is the same longest chain as before. Indeed, in a well-running network (ie. the orphan rate is low), almost all of the time LMD GHOST and the longest chain rule <em>will</em> give the exact same answer. But in more extreme circumstances, this is not always true. For example, consider the following chain, with a more substantial three-block fork:</p>
<center>
<img src="https://vitalik.ca/files/Chain6.png" /><br />
<small><i>Scoring blocks by chain length. If we follow the longest chain rule, the top chain is longer, so the top chain wins.</i></small>
</center>
<p><br /></p>
<center>
<img src="https://vitalik.ca/files/Chain7.png" /><br />
<small><i>Scoring blocks by number of supporting latest messages and using the GHOST rule (latest message from each validator shown in blue). The bottom chain has more recent support, so if we follow the LMD GHOST rule the bottom chain wins, though it's not yet clear which of the three blocks takes precedence.</i></small>
</center>
<p><br /></p>
<p>The LMD GHOST approach is advantageous in part because it is better at extracting information in conditions of high latency. If two validators create two blocks with the same parent, they should really be both counted as cooperating votes for the parent block, even though they are at the same time competing votes for themselves. The longest chain rule fails to capture this nuance; GHOST-based rules do.</p>
<h3 id="detecting-finality">Detecting finality</h3>
<p>But the LMD GHOST approach has another nice property: it’s <em>sticky</em>. For example, suppose that for two rounds, 4/5 of validators voted for the same chain (we’ll assume that the one of the five validators that did not, B, is attacking):</p>
<center>
<img src="https://vitalik.ca/files/Chain8.png" /><br />
</center>
<p><br /></p>
<p>What would need to actually happen for the chain on top to become the canonical chain? Four of five validators built on top of E’s first block, and all four recognized that E had a high score in the LMD fork choice. Just by looking at the structure of the chain, we can know for a fact at least some of the messages that the validators must have seen at different times. Here is what we know about the four validators’ views:</p>
<center>
<table style="text-align:center" cellpadding="20px"><tr>
<td><img src="https://vitalik.ca/files/Chain9.png" width="300px" /><br /><i>A's view</i></td>
<td><img src="https://vitalik.ca/files/Chain10.png" width="300px" /><br /><i>C's view</i></td>
</tr><tr>
<td><img src="https://vitalik.ca/files/Chain11.png" width="300px" /><br /><i>D's view</i></td>
<td><img src="https://vitalik.ca/files/Chain11point5.png" width="300px" /><br /><i>E's view</i></td>
</tr></table>
<small><i>Blocks produced by each validator in green, the latest messages we know that they saw from each of the other validators in blue.</i></small>
</center>
<p><br /></p>
<p>Note that all four of the validators <em>could have</em> seen one or both of B’s blocks, and D and E <em>could have</em> seen C’s second block, making that the latest message in their views instead of C’s first block; however, the structure of the chain itself gives us no evidence that they actually did. Fortunately, as we will see below, this ambiguity does not matter for us.</p>
<p>A’s view contains four latest-messages supporting the bottom chain, and none supporting B’s block. Hence, in (our simulation of) A’s eyes the score in favor of the bottom chain is <em>at least</em> 4-1. The views of C, D and E paint a similar picture, with four latest-messages supporting the bottom chain. Hence, all four of the validators are in a position where they cannot change their minds unless two other validators change their minds first to bring the score to 2-3 in favor of B’s block.</p>
<p>Note that our simulation of the validators’ views is “out of date” in that, for example, it does not capture that D and E could have seen the more recent block by C. However, this does not alter the calculation for the top vs bottom chain, because we can very generally say that any validator’s new message will have the same opinion as their previous messages, unless two other validators have already switched sides first.</p>
<center>
<img src="https://vitalik.ca/files/Chain12.png" width="700px" /><br />
<small><i>A minimal viable attack. A and C illegally switch over to support B's block (and can get penalized for this), giving it a 3-2 advantage, and at this point it becomes legal for D and E to also switch over.</i></small>
</center>
<p><br /></p>
<p>Since fork choice rules such as LMD GHOST are sticky in this way, and clients can detect when the fork choice rule is “stuck on” a particular block, we can use this as a way of achieving asynchronously safe consensus.</p>
<h3 id="safety-oracles">Safety Oracles</h3>
<p>Actually detecting all possible situations where the chain becomes stuck on some block (in CBC lingo, the block is “decided” or “safe”) is very difficult, but we can come up with a set of heuristics (“safety oracles”) which will help us detect <em>some</em> of the cases where this happens. The simplest of these is the <strong>clique oracle</strong>. If there exists some subset <code class="highlighter-rouge">V</code> of the validators making up portion <code class="highlighter-rouge">p</code> of the total validator set (with <code class="highlighter-rouge">p > 1/2</code>) that all make blocks supporting some block <code class="highlighter-rouge">B</code> and then make another round of blocks still supporting <code class="highlighter-rouge">B</code> that references their first round of blocks, then we can reason as follows:</p>
<p>Because of the two rounds of messaging, we know that this subset <code class="highlighter-rouge">V</code> all (i) support <code class="highlighter-rouge">B</code> (ii) know that <code class="highlighter-rouge">B</code> is well-supported, and so none of them can legally switch over unless enough others switch over first. For some competing <code class="highlighter-rouge">B'</code> to beat out <code class="highlighter-rouge">B</code>, the support such a <code class="highlighter-rouge">B'</code> can <em>legally</em> have is initially at most <code class="highlighter-rouge">1-p</code> (everyone not part of the clique), and to win the LMD GHOST fork choice its support needs to get to <code class="highlighter-rouge">1/2</code>, so at least <code class="highlighter-rouge">1/2 - (1-p) = p - 1/2</code> need to illegally switch over to get it to the point where the LMD GHOST rule supports <code class="highlighter-rouge">B'</code>.</p>
<p>As a specific case, note that the <code class="highlighter-rouge">p=3/4</code> clique oracle offers a <code class="highlighter-rouge">1/4</code> level of safety, and a set of blocks satisfying the clique can (and in normal operation, will) be generated as long as <code class="highlighter-rouge">3/4</code> of nodes are online. Hence, in a BFT sense, the level of fault tolerance that can be reached using two-round clique oracles is <code class="highlighter-rouge">1/4</code>, in terms of both liveness and safety.</p>
<p>This approach to consensus has many nice benefits. First of all, the short-term chain selection algorithm, and the “finality algorithm”, are not two awkwardly glued together distinct components, as they admittedly are in Casper FFG; rather, they are both part of the same coherent whole. Second, because safety detection is client-side, there is no need to choose any thresholds in-protocol; clients can decide for themselves what level of safety is sufficient to consider a block as finalized.</p>
<h3 id="going-further">Going Further</h3>
<p>CBC can be extended further in many ways. First, one can come up with other safety oracles; higher-round clique oracles can reach <code class="highlighter-rouge">1/3</code> fault tolerance. Second, we can add validator rotation mechanisms. The simplest is to allow the validator set to change by a small percentage every time the <code class="highlighter-rouge">q=3/4</code> clique oracle is satisfied, but there are other things that we can do as well. Third, we can go beyond chain-like structures, and instead look at structures that increase the density of messages per unit time, like the Serenity beacon chain’s attestation structure:</p>
<center>
<img src="https://vitalik.ca/files/Chain13.png" /><br />
</center>
<p><br /></p>
<p>In this case, it becomes worthwhile to separate <em>attestations</em> from <em>blocks</em>; a block is an object that actually grows the underlying DAG, whereas an attestation contributes to the fork choice rule. In the <a href="http://github.com/ethereum/eth2.0-specs">Serenity beacon chain spec</a>, each block may have hundreds of attestations corresponding to it. However, regardless of which way you do it, the core logic of CBC Casper remains the same.</p>
<p>To make CBC Casper’s safety “cryptoeconomically enforceable”, we need to add validity and slashing conditions. First, we’ll start with the validity rule. A block contains both a parent block and a set of attestations that it knows about that are not yet part of the chain (similar to “uncles” in the current Ethereum PoW chain). For the block to be valid, the block’s parent must be the result of executing the LMD GHOST fork choice rule given the information included in the chain including in the block itself.</p>
<center>
<img src="https://vitalik.ca/files/Chain14.png" /><br />
<small><i>Dotted lines are uncle links, eg. when E creates a block, E notices that C is not yet part of the chain, and so includes a reference to C.</i></small>
</center>
<p><br /></p>
<p>We now can make CBC Casper safe with only one slashing condition: you cannot make two attestations M1 and M2, unless either M1 is in the chain that M2 is attesting to or M2 is in the chain that M1 is attesting to.</p>
<center>
<table style="text-align:center" cellpadding="20px"><tr>
<td><img src="https://vitalik.ca/files/Chain15.png" width="280px" /><br />OK</td>
<td><img src="https://vitalik.ca/files/Chain16.png" width="280px" /><br />Not OK</td>
</tr></table>
</center>
<p>The validity and slashing conditions are relatively easy to describe, though actually implementing them requires checking hash chains and executing fork choice rules in-consensus, so it is not nearly as simple as taking two messages and checking a couple of inequalities between the numbers that these messages commit to, as you can do in Casper FFG for the <code class="highlighter-rouge">NO_SURROUND</code> and <code class="highlighter-rouge">NO_DBL_VOTE</code> <a href="https://ethresear.ch/t/beacon-chain-casper-ffg-rpj-mini-spec/2760">slashing conditions</a>.</p>
<p>Liveness in CBC Casper piggybacks off of the liveness of whatever the underlying chain algorithm is (eg. if it’s one-block-per-slot, then it depends on a synchrony assumption that all nodes will see everything produced in slot N before the start of slot N+1). It’s not possible to get “stuck” in such a way that one cannot make progress; it’s possible to get to the point of finalizing new blocks from any situation, even one where there are attackers and/or network latency is higher than that required by the underlying chain algorithm.</p>
<p>Suppose that at some time T, the network “calms down” and synchrony assumptions are once again satisfied. Then, everyone will converge on the same view of the chain, with the same head H. From there, validators will begin to sign messages supporting H or descendants of H. From there, the chain can proceed smoothly, and will eventually satisfy a clique oracle, at which point H becomes finalized.</p>
<center>
<img src="https://vitalik.ca/files/Chain17.png" height="100px" /><br />
<small><i>Chaotic network due to high latency.</i></small>
</center>
<p><br /></p>
<center>
<img src="https://vitalik.ca/files/Chain18.png" height="100px" /><br />
<small><i>Network latency subsides, a majority of validators see all of the same blocks or at least enough of them to get to the same head when executing the fork choice, and start building on the head, further reinforcing its advantage in the fork choice rule.</i></small>
</center>
<p><br /></p>
<center>
<img src="https://vitalik.ca/files/Chain19.png" height="100px" /><br />
<small><i>Chain proceeds "peacefully" at low latency. Soon, a clique oracle will be satisfied.</i></small>
</center>
<p><br /></p>
<p>That’s all there is to it! Implementation-wise, CBC may arguably be considerably more complex than FFG, but in terms of ability to reason about the protocol, and the properties that it provides, it’s surprisingly simple.</p>
Wed, 05 Dec 2018 17:03:10 -0800
https://vitalik.ca/general/2018/12/05/cbc_casper.html
https://vitalik.ca/general/2018/12/05/cbc_casper.htmlgeneralLayer 1 Should Be Innovative in the Short Term but Less in the Long Term<p><strong>See update 2018-08-29</strong></p>
<p>One of the key tradeoffs in blockchain design is whether to build more functionality into base-layer blockchains themselves (“layer 1”), or to build it into protocols that live on top of the blockchain, and can be created and modified without changing the blockchain itself (“layer 2”). The tradeoff has so far shown itself most in the scaling debates, with block size increases (and <a href="https://github.com/ethereum/wiki/wiki/Sharding-FAQ">sharding</a>) on one side and layer-2 solutions like Plasma and channels on the other, and to some extent blockchain governance, with loss and theft recovery being solvable by either <a href="https://qz.com/730004/everything-you-need-to-know-about-the-ethereum-hard-fork/">the DAO fork</a> or generalizations thereof such as <a href="https://github.com/ethereum/EIPs/blob/master/EIPS/eip-867.md">EIP 867</a>, or by layer-2 solutions such as <a href="https://www.reddit.com/r/MakerDAO/comments/8fmks1/introducing_reversible_eth_reth_never_send_ether/">Reversible Ether (RETH)</a>. So which approach is ultimately better? Those who know me well, or have seen me <a href="https://twitter.com/VitalikButerin/status/1032589339367231488">out myself as a dirty centrist</a>, know that I will inevitably say “some of both”. However, in the longer term, I do think that as blockchains become more and more mature, layer 1 will necessarily stabilize, and layer 2 will take on more and more of the burden of ongoing innovation and change.</p>
<p>There are several reasons why. The first is that layer 1 solutions require ongoing protocol change to happen at the base protocol layer, base layer protocol change requires governance, and <strong>it has still not been shown that, in the long term, highly “activist” blockchain governance can continue without causing ongoing political uncertainty or collapsing into centralization</strong>.</p>
<p>To take an example from another sphere, consider Moxie Marlinspike’s <a href="https://signal.org/blog/the-ecosystem-is-moving/">defense of Signal’s centralized and non-federated nature</a>. A document by a company defending its right to maintain control over an ecosystem it depends on for its key business should of course be viewed with massive grains of salt, but one can still benefit from the arguments. Quoting:</p>
<blockquote>
<p>One of the controversial things we did with Signal early on was to build it as an unfederated service. Nothing about any of the protocols we’ve developed requires centralization; it’s entirely possible to build a federated Signal Protocol-based messenger, but I no longer believe that it is possible to build a competitive federated messenger at all.</p>
</blockquote>
<p>And:</p>
<blockquote>
<p>Their retort was “that’s dumb, how far would the internet have gotten without interoperable protocols defined by 3rd parties?”
I thought about it. We got to the first production version of IP, and have been trying for the past 20 years to switch to a second production version of IP with limited success. We got to HTTP version 1.1 in 1997, and have been stuck there until now. Likewise, SMTP, IRC, DNS, XMPP, are all similarly frozen in time circa the late 1990s. To answer his question, that’s how far the internet got. It got to the late 90s.<br />
That has taken us pretty far, but it’s undeniable that once you federate your protocol, it becomes very difficult to make changes. And right now, at the application level, things that stand still don’t fare very well in a world where the ecosystem is moving …
So long as federation means stasis while centralization means movement, federated protocols are going to have trouble existing in a software climate that demands movement as it does today.</p>
</blockquote>
<p>At this point in time, and in the medium term going forward, it seems clear that decentralized application platforms, cryptocurrency payments, identity systems, reputation systems, decentralized exchange mechanisms, auctions, privacy solutions, programming languages that support privacy solutions, and most other interesting things that can be done on blockchains are spheres where there will continue to be significant and ongoing innovation. Decentralized application platforms often need continued reductions in confirmation time, payments need fast confirmations, low transaction costs, privacy, and many other built-in features, exchanges are appearing in many shapes and sizes including <a href="https://uniswap.io/">on-chain automated market makers</a>, <a href="https://www.cftc.gov/sites/default/files/idc/groups/public/@newsroom/documents/file/tac021014_budish.pdf">frequent batch auctions</a>, <a href="http://cramton.umd.edu/ca-book/cramton-shoham-steinberg-combinatorial-auctions.pdf">combinatorial auctions</a> and more. Hence, “building in” any of these into a base layer blockchain would be a bad idea, as it would create a high level of governance overhead as the platform would have to continually discuss, implement and coordinate newly discovered technical improvements. For the same reason federated messengers have a hard time getting off the ground without re-centralizing, blockchains would also need to choose between adopting activist governance, with the perils that entails, and falling behind newly appearing alternatives.</p>
<p>Even Ethereum’s limited level of application-specific functionality, precompiles, has seen some of this effect. Less than a year ago, Ethereum adopted the Byzantium hard fork, including operations to facilitate <a href="https://github.com/ethereum/EIPs/blob/master/EIPS/eip-196.md">elliptic curve</a> <a href="https://github.com/ethereum/EIPs/blob/master/EIPS/eip-197.md">operations</a> needed for ring signatures, ZK-SNARKs and other applications, using the <a href="https://github.com/topics/alt-bn128">alt-bn128</a> curve. Now, Zcash and other blockchains are moving toward <a href="https://blog.z.cash/new-snark-curve/">BLS-12-381</a>, and Ethereum would need to fork again to catch up. In part to avoid having similar problems in the future, the Ethereum community is looking to upgrade the EVM to <a href="https://github.com/ewasm/design">E-WASM</a>, a virtual machine that is sufficiently more efficient that there is far less need to incorporate application-specific precompiles.</p>
<p>But there is also a second argument in favor of layer 2 solutions, one that does not depend on speed of anticipated technical development: <em>sometimes there are inevitable tradeoffs, with no single globally optimal solution</em>. This is less easily visible in Ethereum 1.0-style blockchains, where there are certain models that are reasonably universal (eg. Ethereum’s account-based model is one). In <em>sharded</em> blockchains, however, one type of question that does <em>not</em> exist in Ethereum today crops up: how to do cross-shard transactions? That is, suppose that the blockchain state has regions A and B, where few or no nodes are processing both A and B. How does the system handle transactions that affect both A and B?</p>
<p>The <a href="https://github.com/ethereum/wiki/wiki/Sharding-FAQs#how-can-we-facilitate-cross-shard-communication">current answer</a> involves asynchronous cross-shard communication, which is sufficient for transferring assets and some other applications, but insufficient for many others. Synchronous operations (eg. to solve the <a href="https://github.com/ethereum/wiki/wiki/Sharding-FAQs#what-is-the-train-and-hotel-problem">train and hotel problem</a>) can be bolted on top with <a href="https://ethresear.ch/t/cross-shard-contract-yanking/1450">cross-shard yanking</a>, but this requires multiple rounds of cross-shard interaction, leading to significant delays. We can solve these problems with a <a href="https://ethresear.ch/t/simple-synchronous-cross-shard-transaction-protocol/3097">synchronous execution scheme</a>, but this comes with several tradeoffs:</p>
<ul>
<li>The system cannot process more than one transaction for the same account per block</li>
<li>Transactions must declare in advance what shards and addresses they affect</li>
<li>There is a high risk of any given transaction failing (and still being required to pay fees!) if the transaction is only accepted in some of the shards that it affects but not others</li>
</ul>
<p>It seems very likely that a better scheme can be developed, but it would be more complex, and may well have limitations that this scheme does not. There are known results preventing perfection; at the very least, <a href="https://en.wikipedia.org/wiki/Amdahl%27s_law">Amdahl’s law</a> puts a hard limit on the ability of some applications and some types of interaction to process more transactions per second through parallelization.</p>
<p>So how do we create an environment where better schemes can be tested and deployed? The answer is an idea that can be credited to Justin Drake: layer 2 execution engines. Users would be able to send assets into a “bridge contract”, which would calculate (using some indirect technique such as <a href="https://truebit.io/">interactive verification</a> or <a href="https://medium.com/@VitalikButerin/zk-snarks-under-the-hood-b33151a013f6">ZK-SNARKs</a>) state roots using some alternative set of rules for processing the blockchain (think of this as equivalent to layer-two “meta-protocols” like <a href="https://blog.omni.foundation/2013/11/29/a-brief-history-of-mastercoin/">Mastercoin/OMNI</a> and <a href="https://counterparty.io/">Counterparty</a> on top of Bitcoin, except because of the bridge contract these protocols would be able to handle assets whose “base ledger” is defined on the underlying protocol), and which would process withdrawals if and only if the alternative ruleset generates a withdrawal request.</p>
<p><br /></p>
<center>
<img src="https://vitalik.ca/files/Layer2.png" />
</center>
<p><br /><br /></p>
<p>Note that anyone can create a layer 2 execution engine at any time, different users can use different execution engines, and one can switch from one execution engine to any other, or to the base protocol, fairly quickly. The base blockchain no longer has to worry about being an optimal smart contract processing engine; it need only be a data availability layer with execution rules that are quasi-Turing-complete so that any layer 2 bridge contract can be built on top, and that allow basic operations to carry state between shards (in fact, only ETH transfers being fungible across shards is sufficient, but it takes very little effort to also allow cross-shard calls, so we may as well support them), but does not require complexity beyond that. Note also that layer 2 execution engines can have different state management rules than layer 1, eg. not having storage rent; anything goes, as it’s the responsibility of the users of that specific execution engine to make sure that it is sustainable, and if they fail to do so the consequences are contained to within the users of that particular execution engine.</p>
<p>In the long run, layer 1 would not be actively competing on all of these improvements; it would simply provide a stable platform for the layer 2 innovation to happen on top. <strong>Does this mean that, say, sharding is a bad idea, and we should keep the blockchain size and state small so that even 10 year old computers can process everyone’s transactions? Absolutely not.</strong> Even if execution engines are something that gets partially or fully moved to layer 2, consensus on data ordering and availability is still a highly generalizable and necessary function; to see how difficult layer 2 execution engines are without layer 1 scalable data availability consensus, <a href="https://ethresear.ch/t/minimal-viable-plasma/426">see</a> the <a href="https://ethresear.ch/t/plasma-cash-plasma-with-much-less-per-user-data-checking/1298">difficulties</a> in <a href="https://ethresear.ch/t/plasma-debit-arbitrary-denomination-payments-in-plasma-cash/2198">Plasma</a> research, and its <a href="https://medium.com/@kelvinfichter/why-is-evm-on-plasma-hard-bf2d99c48df7">difficulty</a> of naturally extending to fully general purpose blockchains, for an example. And if people want to throw a hundred megabytes per second of data into a system where they need consensus on availability, then we need a hundred megabytes per second of data availability consensus.</p>
<p>Additionally, layer 1 can still improve on reducing latency; if layer 1 is slow, the only strategy for achieving very low latency is <a href="https://medium.com/statechannels/counterfactual-generalized-state-channels-on-ethereum-d38a36d25fc6">state channels</a>, which often have high capital requirements and can be difficult to generalize. State channels will always beat layer 1 blockchains in latency as state channels require only a single network message, but in those cases where state channels do not work well, layer 1 blockchains can still come closer than they do today.</p>
<p>Hence, the other extreme position, that blockchain base layers can be truly absolutely minimal, and not bother with either a quasi-Turing-complete execution engine or scalability to beyond the capacity of a single node, is also clearly false; there is a certain minimal level of complexity that is required for base layers to be powerful enough for applications to build on top of them, and we have not yet reached that level. Additional complexity is needed, though it should be chosen very carefully to make sure that it is maximally general purpose, and not targeted toward specific applications or technologies that will go out of fashion in two years due to loss of interest or better alternatives.</p>
<p>And even in the future base layers will need to continue to make some upgrades, especially if new technologies (eg. STARKs reaching higher levels of maturity) allow them to achieve stronger properties than they could before, though developers today can take care to make base layer platforms maximally forward-compatible with such potential improvements. So it will continue to be true that a balance between layer 1 and layer 2 improvements is needed to continue improving scalability, privacy and versatility, though layer 2 will continue to take up a larger and larger share of the innovation over time.</p>
<p><strong>Update 2018.08.29:</strong> Justin Drake pointed out to me another good reason why some features may be best implemented on layer 1: those features are public goods, and so could not be efficiently or reliably funded with feature-specific use fees, and hence are best paid for by subsidies paid out of issuance or burned transaction fees. One possible example of this is secure random number generation, and another is generation of zero knowledge proofs for more efficient client validation of correctness of various claims about blockchain contents or state.</p>
Sun, 26 Aug 2018 18:03:10 -0700
https://vitalik.ca/general/2018/08/26/layer_1.html
https://vitalik.ca/general/2018/08/26/layer_1.htmlgeneralA Guide to 99% Fault Tolerant Consensus<p><em>Special thanks to Emin Gun Sirer for review</em></p>
<p>We’ve heard for a long time that it’s possible to achieve consensus with 50% fault tolerance in a synchronous network where messages broadcasted by any honest node are guaranteed to be received by all other honest nodes within some known time period (if an attacker has <em>more</em> than 50%, they can perform a “51% attack”, and there’s an analogue of this for any algorithm of this type). We’ve also heard for a long time that if you want to relax the synchrony assumption, and have an algorithm that’s “safe under asynchrony”, the maximum achievable fault tolerance drops to 33% (<a href="http://pmg.csail.mit.edu/papers/osdi99.pdf">PBFT</a>, <a href="https://arxiv.org/abs/1710.09437">Casper FFG</a>, etc all fall into this category). But did you know that if you add <em>even more</em> assumptions (specifically, you require <em>observers</em>, ie. users that are not actively participating in the consensus but care about its output, to also be actively watching the consensus, and not just downloading its output after the fact), you can increase fault tolerance all the way to 99%?</p>
<p>This has in fact been known for a long time; Leslie Lamport’s famous 1982 paper “The Byzantine Generals Problem” (link <a href="https://people.eecs.berkeley.edu/~luca/cs174/byzantine.pdf">here</a>) contains a description of the algorithm. The following will be my attempt to describe and reformulate the algorithm in a simplified form.</p>
<p>Suppose that there are <code class="highlighter-rouge">N</code> consensus-participating nodes, and everyone agrees who these nodes are ahead of time (depending on context, they could have been selected by a trusted party or, if stronger decentralization is desired, by some proof of work or proof of stake scheme). We label these nodes <code class="highlighter-rouge">0....N-1</code>. Suppose also that there is a known bound <code class="highlighter-rouge">D</code> on network latency plus clock disparity (eg. <code class="highlighter-rouge">D</code> = 8 seconds). Each node has the ability to publish a value at time <code class="highlighter-rouge">T</code> (a malicious node can of course propose values earlier or later than <code class="highlighter-rouge">T</code>). All nodes wait <code class="highlighter-rouge">(N-1) * D</code> seconds, running the following process. Define <code class="highlighter-rouge">x : i</code> as “the value <code class="highlighter-rouge">x</code> signed by node <code class="highlighter-rouge">i</code>”, <code class="highlighter-rouge">x : i : j</code> as “the value <code class="highlighter-rouge">x</code> signed by <code class="highlighter-rouge">i</code>, and that value and signature together signed by <code class="highlighter-rouge">j</code>”, etc. The proposals published in the first stage will be of the form <code class="highlighter-rouge">v: i</code> for some <code class="highlighter-rouge">v</code> and <code class="highlighter-rouge">i</code>, containing the signature of the node that proposed it.</p>
<p>If a validator <code class="highlighter-rouge">i</code> receives some message <code class="highlighter-rouge">v : i[1] : ... : i[k]</code>, where <code class="highlighter-rouge">i[1] ... i[k]</code> is a list of indices that have (sequentially) signed the message already (just <code class="highlighter-rouge">v</code> by itself would count as k=0, and <code class="highlighter-rouge">v:i</code> as k=1), then the validator checks that (i) the time is less than <code class="highlighter-rouge">T + k * D</code>, and (ii) they have not yet seen a valid message containing <code class="highlighter-rouge">v</code>; if both checks pass, they publish <code class="highlighter-rouge">v : i[1] : ... : i[k] : i</code>.</p>
<p>At time <code class="highlighter-rouge">T + (N-1) * D</code>, nodes stop listening. At this point, there is a guarantee that honest nodes have all “validly seen” the same set of values.</p>
<center>
<img src="http://vitalik.ca/files/Lamport.png" /><br />
<i><small>Node 1 (red) is malicious, and nodes 0 and 2 (grey) are honest. At the start, the two honest nodes make their proposals <code>y</code> and <code>x</code>, and the attacker proposes both <code>w</code> and <code>z</code> late. <code>w</code> reaches node 0 on time but not node 2, and <code>z</code> reaches neither node on time. At time <code>T + D</code>, nodes 0 and 2 rebroadcast all values they've seen that they have not yet broadcasted, but add their signatures on (<code>x</code> and <code>w</code> for node 0, <code>y</code> for node 2). Both honest nodes saw <code>{x, y, w}</code>.</small></i>
</center>
<p><br /></p>
<p>If the problem demands choosing one value, they can use some “choice” function to pick a single value out of the values they have seen (eg. they take the one with the lowest hash). The nodes can then agree on this value.</p>
<p>Now, let’s explore why this works. What we need to prove is that if one honest node has seen a particular value (validly), then every other honest node has also seen that value (and if we prove this, then we know that all honest nodes have seen the same set of values, and so if all honest nodes are running the same choice function, they will choose the same value). Suppose that any honest node receives a message <code class="highlighter-rouge">v : i[1] : ... : i[k]</code> that they perceive to be valid (ie. it arrives before time <code class="highlighter-rouge">T + k * D</code>). Suppose <code class="highlighter-rouge">x</code> is the index of a single other honest node. Either <code class="highlighter-rouge">x</code> is part of <code class="highlighter-rouge">{i[1] ... i[k]}</code> or it is not.</p>
<ul>
<li>In the first case (say <code class="highlighter-rouge">x = i[j]</code> for this message), we know that the honest node <code class="highlighter-rouge">x</code> had already broadcasted that message, and they did so in response to a message with <code class="highlighter-rouge">j-1</code> signatures that they received before time <code class="highlighter-rouge">T + (j-1) * D</code>, so they broadcast their message at that time, and so the message must have been received by all honest nodes before time <code class="highlighter-rouge">T + j * D</code>.</li>
<li>In the second case, since the honest node sees the message before time <code class="highlighter-rouge">T + k * D</code>, then they will broadcast the message with their signature and guarantee that everyone, including <code class="highlighter-rouge">x</code>, will see it before time <code class="highlighter-rouge">T + (k+1) * D</code>.</li>
</ul>
<p>Notice that the algorithm uses the act of adding one’s own signature as a kind of “bump” on the timeout of a message, and it’s this ability that guarantees that if one honest node saw a message on time, they can ensure that everyone else sees the message on time as well, as the definition of “on time” increments by more than network latency with every added signature.</p>
<p>In the case where one node is honest, can we guarantee that passive <em>observers</em> (ie. non-consensus-participating nodes that care about knowing the outcome) can also see the outcome, even if we require them to be watching the process the whole time? With the scheme as written, there’s a problem. Suppose that a commander and some subset of <code class="highlighter-rouge">k</code> (malicious) validators produce a message <code class="highlighter-rouge">v : i[1] : .... : i[k]</code>, and broadcast it directly to some “victims” just before time <code class="highlighter-rouge">T + k * D</code>. The victims see the message as being “on time”, but when they rebroadcast it, it only reaches all honest consensus-participating nodes after <code class="highlighter-rouge">T + k * D</code>, and so all honest consensus-participating nodes reject it.</p>
<center>
<img src="http://vitalik.ca/files/Lamport2.png" />
</center>
<p><br /></p>
<p>But we can plug this hole. We require <code class="highlighter-rouge">D</code> to be a bound on <em>two times</em> network latency plus clock disparity. We then put a different timeout on observers: an observer accepts <code class="highlighter-rouge">v : i[1] : .... : i[k]</code> before time <code class="highlighter-rouge">T + (k - 0.5) * D</code>. Now, suppose an observer sees a message an accepts it. They will be able to broadcast it to an honest node before time <code class="highlighter-rouge">T + k * D</code>, and the honest node will issue the message with their signature attached, which will reach all other observers before time <code class="highlighter-rouge">T + (k + 0.5) * D</code>, the timeout for messages with <code class="highlighter-rouge">k+1</code> signatures.</p>
<center>
<img src="http://vitalik.ca/files/Lamport3.png" />
</center>
<p><br /></p>
<h3 id="retrofitting-onto-other-consensus-algorithms">Retrofitting onto other consensus algorithms</h3>
<p>The above could theoretically be used as a standalone consensus algorithm, and could even be used to run a proof-of-stake blockchain. The validator set of round N+1 of the consensus could itself be decided during round N of the consensus (eg. each round of a consensus could also accept “deposit” and “withdraw” transactions, which if accepted and correctly signed would add or remove validators into the next round). The main additional ingredient that would need to be added is a mechanism for deciding who is allowed to propose blocks (eg. each round could have one designated proposer). It could also be modified to be usable as a proof-of-work blockchain, by allowing consensus-participating nodes to “declare themselves” in real time by publishing a proof of work solution on top of their public key at th same time as signing a message with it.</p>
<p>However, the synchrony assumption is very strong, and so we would like to be able to work without it in the case where we don’t need more than 33% or 50% fault tolerance. There is a way to accomplish this. Suppose that we have some other consensus algorithm (eg. PBFT, Casper FFG, chain-based PoS) whose output <em>can</em> be seen by occasionally-online observers (we’ll call this the <em>threshold-dependent</em> consensus algorithm, as opposed to the algorithm above, which we’ll call the <em>latency-dependent</em> consensus algorithm). Suppose that the threshold-dependent consensus algorithm runs continuously, in a mode where it is constantly “finalizing” new blocks onto a chain (ie. each finalized value points to some previous finalized value as a “parent”; if there’s a sequence of pointers <code class="highlighter-rouge">A -> ... -> B</code>, we’ll call A a <em>descendant</em> of B).</p>
<p>We can retrofit the latency-dependent algorithm onto this structure, giving always-online observers access to a kind of “strong finality” on checkpoints, with fault tolerance ~95% (you can push this arbitrarily close to 100% by adding more validators and requiring the process to take longer).</p>
<p>Every time the time reaches some multiple of 4096 seconds, we run the latency-dependent algorithm, choosing 512 random nodes to participate in the algorithm. A valid proposal is any valid chain of values that were finalized by the threshold-dependent algorithm. If a node sees some finalized value before time <code class="highlighter-rouge">T + k * D</code> (D = 8 seconds) with <code class="highlighter-rouge">k</code> signatures, it accepts the chain into its set of known chains and rebroadcasts it with its own signature added; observers use a threshold of <code class="highlighter-rouge">T + (k - 0.5) * D</code> as before.</p>
<p>The “choice” function used at the end is simple:</p>
<ul>
<li>Finalized values that are not descendants of what was already agreed to be a finalized value in the previous round are ignored</li>
<li>Finalized values that are invalid are ignored</li>
<li>To choose between two valid finalized values, pick the one with the lower hash</li>
</ul>
<p>If 5% of validators are honest, there is only a roughly 1 in 1 trillion chance that none of the 512 randomly selected nodes will be honest, and so as long as the network latency plus clock disparity is less than <code class="highlighter-rouge">D/2</code> the above algorithm will work, correctly coordinating nodes on some single finalized value, even if multiple conflicting finalized values are presented because the fault tolerance of the threshold-dependent algorithm is broken.</p>
<p>If the fault tolerance of the threshold-dependent consensus algorithm is met (usually 50% or 67% honest), then the threshold-dependent consensus algorithm will either not finalize any new checkpoints, or it will finalize new checkpoints that are compatible with each other (eg. a series of checkpoints where each points to the previous as a parent), so even if network latency exceeds <code class="highlighter-rouge">D/2</code> (or even <code class="highlighter-rouge">D</code>), and as a result nodes participating in the latency-dependent algorithm disagree on which value they accept, the values they accept are still guaranteed to be part of the same chain and so there is no actual disagreement. Once latency recovers back to normal in some future round, the latency-dependent consensus will get back “in sync”.</p>
<p>If the assumptions of both the threshold-dependent and latency-dependent consensus algorithms are broken <em>at the same time</em> (or in consecutive rounds), then the algorithm can break down. For example, suppose in one round, the threshold-dependent consensus finalizes <code class="highlighter-rouge">Z -> Y -> X</code> and the latency-dependent consensus disagrees between <code class="highlighter-rouge">Y</code> and <code class="highlighter-rouge">X</code>, and in the next round the threshold-dependent consensus finalizes a descendant <code class="highlighter-rouge">W</code> of <code class="highlighter-rouge">X</code> which is <em>not</em> a descendant of <code class="highlighter-rouge">Y</code>; in the latency-dependent consensus, the nodes who agreed <code class="highlighter-rouge">Y</code> will not accept <code class="highlighter-rouge">W</code>, but the nodes that agreed <code class="highlighter-rouge">X</code> will. However, this is unavoidable; the impossibility of safe-under-asynchrony consensus with more than 1/3 fault tolerance is a <a href="https://groups.csail.mit.edu/tds/papers/Lynch/jacm88.pdf">well known result</a> in Byzantine fault tolerance theory, as is the impossibility of more than 1/2 fault tolerance even allowing synchrony assumptions but assuming offline observers.</p>
Tue, 07 Aug 2018 18:03:10 -0700
https://vitalik.ca/general/2018/08/07/99_fault_tolerant.html
https://vitalik.ca/general/2018/08/07/99_fault_tolerant.htmlgeneralSTARKs, Part 3: Into the Weeds<p><em>Special thanks to Eli ben Sasson for his kind assistance, as usual. Special thanks to Chih-Cheng Liang and Justin Drake for review, and to Ben Fisch for suggesting the reverse MIMC technique for a VDF (paper <a href="https://eprint.iacr.org/2018/601.pdf">here</a>)</em></p>
<p><em>Trigger warning: math and lots of python</em></p>
<style>
div.foo {
color: white;
}
div.foo:hover {
color: black;
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<p>As a followup to <a href="https://vitalik.ca/general/2017/11/09/starks_part_1.html">Part 1</a> and <a href="https://vitalik.ca/general/2017/11/22/starks_part_2.html">Part 2</a> of this series, this post will cover what it looks like to actually implement a STARK, complete with an implementation in python. STARKs (“Scalable Transparent ARgument of Knowledge” are a technique for creating a proof that <code class="highlighter-rouge">f(x)=y</code> where <code class="highlighter-rouge">f</code> may potentially take a very long time to calculate, but where the proof can be verified very quickly. A STARK is “doubly scalable”: for a computation with <code class="highlighter-rouge">t</code> steps, it takes roughly <code class="highlighter-rouge">O(t * log(t))</code> steps to produce a proof, which is likely optimal, and it takes <code class="highlighter-rouge">~O(log</code><sup><code class="highlighter-rouge">2</code></sup><code class="highlighter-rouge">(t))</code> steps to verify, which for even moderately large values of <code class="highlighter-rouge">t</code> is much faster than the original computation. STARKs can also have a privacy-preserving “zero knowledge” property, though the use case we will apply them to here, making verifiable delay functions, does not require this property, so we do not need to worry about it.</p>
<p>First, some disclaimers:</p>
<ul>
<li>This code has not been thoroughly audited; soundness in production use cases is not guaranteed</li>
<li>This code is very suboptimal (it’s written in Python, what did you expect)</li>
<li>STARKs “in real life” (ie. as implemented in Eli and co’s production implementations) tend to use binary fields and not prime fields for application-specific efficiency reasons; however, they do stress in their writings the prime field-based approach to STARKs described here is legitimate and can be used</li>
<li>There is no “one true way” to do a STARK. It’s a broad category of cryptographic and mathematical constructs, with different setups optimal for different applications and constant ongoing research to reduce prover and verifier complexity and improve soundness.</li>
<li>This article absolutely expects you to know how modular arithmetic and prime fields work, and be comfortable with the concepts of polynomials, interpolation and evaluation. If you don’t, go back to <a href="https://vitalik.ca/general/2017/11/22/starks_part_2.html">Part 2</a>, and also this <a href="https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649">earlier post on quadratic arithmetic programs</a></li>
</ul>
<p>Now, let’s get to it.</p>
<h3 id="mimc">MIMC</h3>
<p>Here is the function we’ll be doing a STARK of:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>def mimc(inp, steps, round_constants):
start_time = time.time()
for i in range(steps-1):
inp = (inp**3 + round_constants[i % len(round_constants)]) % modulus
print("MIMC computed in %.4f sec" % (time.time() - start_time))
return inp
</code></pre></div></div>
<p>We choose MIMC (see <a href="https://eprint.iacr.org/2016/492.pdf">paper</a>) as the example because it is both (i) simple to understand and (ii) interesting enough to be useful in real life. The function can be viewed visually as follows:</p>
<center>
<img src="http://vitalik.ca/files/MIMC.png" /><br />
<br />
<small><i>Note: in many discussions of MIMC, you will typically see XOR used instead of +; this is because MIMC is typically done over binary fields, where addition _is_ XOR; here we are doing it over prime fields.</i></small>
</center>
<p>In our example, the round constants will be a relatively small list (eg. 64 items) that gets cycled through over and over again (that is, after k[64] it loops back to using k[1]).</p>
<p>MIMC with a very large number of rounds, as we’re doing here, is useful as a <em>verifiable delay function</em> - a function which is difficult to compute, and particularly non-parallelizable to compute, but relatively easy to verify. MIMC by itself achieves this property to some extent because MIMC <em>can</em> be computed “backward” (recovering the “input” from its corresponding “output”), but computing it backward takes about 100 times longer to compute than the forward direction (and neither direction can be significantly sped up by parallelization). So you can think of computing the function in the backward direction as being the act of “computing” the non-parallelizable proof of work, and computing the function in the forward direction as being the process of “verifying” it.</p>
<center>
<img src="http://vitalik.ca/files/MIMC2.png" /><br />
<br />
<small><i>x -> x<sup>(2p-1)/3</sup> gives the inverse of x -> x<sup>3</sup>; this is true because of <a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem">Fermat's Little Theorem</a>, a theorem that despite its supposed littleness is arguably much more important to mathematics than Fermat's more famous "Last Theorem".</i></small>
</center>
<p>What we will try to achieve here is to make verification much more efficient by using a STARK - instead of the verifier having to run MIMC in the forward direction themselves, the prover, after completing the computation in the “backward direction”, would compute a STARK of the computation in the “forward direction”, and the verifier would simply verify the STARK. The hope is that the overhead of computing a STARK can be less than the difference in speed running MIMC forwards relative to backwards, so a prover’s time would still be dominated by the initial “backward” computation, and not the (highly parallelizable) STARK computation. Verification of a STARK can be relatively fast (in our python implementation, ~0.05-0.3 seconds), no matter how long the original computation is.</p>
<p>All calculations are done modulo 2<sup>256</sup> - 351 * 2<sup>32</sup> + 1; we are using this prime field modulus because it is the largest prime below 2<sup>256</sup> whose multiplicative group contains an order 2<sup>32</sup> subgroup (that is, there’s a number <code class="highlighter-rouge">g</code> such that successive powers of <code class="highlighter-rouge">g</code> modulo this prime loop around back to 1 after exactly 2<sup>32</sup> cycles), and which is of the form <code class="highlighter-rouge">6k+5</code>. The first property is necessary to make sure that our efficient versions of the FFT and FRI algorithms can work, and the second ensures that MIMC actually can be computed “backwards” (see the use of x -> x<sup>(2p-1)/3</sup> above).</p>
<h3 id="prime-field-operations">Prime field operations</h3>
<p>We start off by building a convenience class that does prime field operations, as well as operations with polynomials over prime fields. The code is <a href="https://github.com/ethereum/research/blob/master/mimc_stark/poly_utils.py">here</a>. First some trivial bits:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>class PrimeField():
def __init__(self, modulus):
# Quick primality test
assert pow(2, modulus, modulus) == 2
self.modulus = modulus
def add(self, x, y):
return (x+y) % self.modulus
def sub(self, x, y):
return (x-y) % self.modulus
def mul(self, x, y):
return (x*y) % self.modulus
</code></pre></div></div>
<p>And the <a href="https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm">Extended Euclidean Algorithm</a> for computing modular inverses (the equivalent of computing 1/x in a prime field):</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Modular inverse using the extended Euclidean algorithm
def inv(self, a):
if a == 0:
return 0
lm, hm = 1, 0
low, high = a % self.modulus, self.modulus
while low > 1:
r = high//low
nm, new = hm-lm*r, high-low*r
lm, low, hm, high = nm, new, lm, low
return lm % self.modulus
</code></pre></div></div>
<p>The above algorithm is relatively expensive; fortunately, for the special case where we need to do many modular inverses, there’s a simple mathematical trick that allows us to compute many inverses, called <a href="https://books.google.com/books?id=kGu4lTznRdgC&pg=PA54&lpg=PA54&dq=montgomery+batch+inversion&source=bl&ots=tPJcPPOrCe&sig=Z3p_6YYwYloRU-f1K-nnv2D8lGw&hl=en&sa=X&ved=0ahUKEwjO8sumgJjcAhUDd6wKHWGNA9cQ6AEIRDAE#v=onepage&q=montgomery%20batch%20inversion&f=false">Montgomery batch inversion</a>:</p>
<center>
<img src="http://vitalik.ca/files/MultiInv.png" /><br />
<br />
<small><i>Using Montgomery batch inversion to compute modular inverses. Inputs purple, outputs green, multiplication gates black; the red square is the _only_ modular inversion.</i></small>
</center>
<p>The code below implements this algorithm, with some slightly ugly special case logic so that if there are zeroes in the set of what we are inverting, it sets their inverse to 0 and moves along.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>def multi_inv(self, values):
partials = [1]
for i in range(len(values)):
partials.append(self.mul(partials[-1], values[i] or 1))
inv = self.inv(partials[-1])
outputs = [0] * len(values)
for i in range(len(values), 0, -1):
outputs[i-1] = self.mul(partials[i-1], inv) if values[i-1] else 0
inv = self.mul(inv, values[i-1] or 1)
return outputs
</code></pre></div></div>
<p>This batch inverse algorithm will prove important later on, when we start dealing with dividing sets of evaluations of polynomials.</p>
<p>Now we move on to some polynomial operations. We treat a polynomial as an array, where element i is the ith degree term (eg. x<sup>3</sup> + 2x + 1 becomes <code class="highlighter-rouge">[1, 2, 0, 1]</code>). Here’s the operation of evaluating a polynomial at <em>one point</em>:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Evaluate a polynomial at a point
def eval_poly_at(self, p, x):
y = 0
power_of_x = 1
for i, p_coeff in enumerate(p):
y += power_of_x * p_coeff
power_of_x = (power_of_x * x) % self.modulus
return y % self.modulus
</code></pre></div></div>
<p><br /></p>
<blockquote><b>Challenge</b><br />
What is the output of <code>f.eval_poly_at([4, 5, 6], 2)</code> if the modulus is 31?<br />
<br />
<b>Mouseover below for answer</b>
<br />
<div class="foo">
6 * 2<sup>2</sup> + 5 * 2 + 4 = 38, 38 mod 31 = 7.
</div>
</blockquote>
<p>There is also code for adding, subtracting, multiplying and dividing polynomials; this is textbook long addition/subtraction/multiplication/division. The one non-trivial thing is Lagrange interpolation, which takes as input a set of x and y coordinates, and returns the minimal polynomial that passes through all of those points (you can think of it as being the inverse of polynomial evaluation):</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Build a polynomial that returns 0 at all specified xs
def zpoly(self, xs):
root = [1]
for x in xs:
root.insert(0, 0)
for j in range(len(root)-1):
root[j] -= root[j+1] * x
return [x % self.modulus for x in root]
def lagrange_interp(self, xs, ys):
# Generate master numerator polynomial, eg. (x - x1) * (x - x2) * ... * (x - xn)
root = self.zpoly(xs)
# Generate per-value numerator polynomials, eg. for x=x2,
# (x - x1) * (x - x3) * ... * (x - xn), by dividing the master
# polynomial back by each x coordinate
nums = [self.div_polys(root, [-x, 1]) for x in xs]
# Generate denominators by evaluating numerator polys at each x
denoms = [self.eval_poly_at(nums[i], xs[i]) for i in range(len(xs))]
invdenoms = self.multi_inv(denoms)
# Generate output polynomial, which is the sum of the per-value numerator
# polynomials rescaled to have the right y values
b = [0 for y in ys]
for i in range(len(xs)):
yslice = self.mul(ys[i], invdenoms[i])
for j in range(len(ys)):
if nums[i][j] and ys[i]:
b[j] += nums[i][j] * yslice
return [x % self.modulus for x in b]
</code></pre></div></div>
<p>See <a href="https://blog.ethereum.org/2014/08/16/secret-sharing-erasure-coding-guide-aspiring-dropbox-decentralizer/">the “M of N” section of this article</a> for a description of the math. Note that we also have special-case methods <code class="highlighter-rouge">lagrange_interp_4</code> and <code class="highlighter-rouge">lagrange_interp_2</code> to speed up the very frequent operations of Lagrange interpolation of degree < 2 and degree < 4 polynomials.</p>
<h3 id="fast-fourier-transforms">Fast Fourier Transforms</h3>
<p>If you read the above algorithms carefully, you might notice that Lagrange interpolation and multi-point evaluation (that is, evaluating a degree < N polynomial at N points) both take quadratic time to execute, so for example doing a Lagrange interpolation of one thousand points takes a few million steps to execute, and a Lagrange interpolation of one million points takes a few trillion. This is an unacceptably high level of inefficiency, so we will use a more efficient algorithm, the Fast Fourier Transform.</p>
<p>The FFT only takes <code class="highlighter-rouge">O(n * log(n))</code> time (ie. ~10,000 steps for 1,000 points, ~20 million steps for 1 million points), though it is more restricted in scope; the x coordinates must be a complete set of <strong><a href="https://en.wikipedia.org/wiki/Root_of_unity">roots of unity</a></strong> of some <strong><a href="https://en.wikipedia.org/wiki/Order_(group_theory)">order</a></strong> <code class="highlighter-rouge">N = 2</code><sup><code class="highlighter-rouge">k</code></sup>. That is, if there are <code class="highlighter-rouge">N</code> points, the x coordinates must be successive powers 1, p, p<sup>2</sup>, p<sup>3</sup>… of some <code class="highlighter-rouge">p</code> where p<sup>N</sup> = 1. The algorithm can, surprisingly enough, be used for multi-point evaluation <em>or</em> interpolation, with one small parameter tweak.</p>
<p><br /></p>
<blockquote><b>Challenge</b>
Find a 16th root of unity mod 337 that is not an 8th root of unity.<br />
<br />
<b>Mouseover below for answer</b>
<br />
<div class="foo">
<code style="background-color:white">59, 146, 30, 297, 278, 191, 307, 40</code><br />
<br />
You could have gotten this list by doing something like <code style="background-color:white">[print(x) for x in range(337) if pow(x, 16, 337) == 1 and pow(x, 8, 337) != 1]</code>, though there is a smarter way that works for much larger moduluses: first, identify a single <i>primitive root</i> mod 337 (that is, not a perfect square), by looking for a value <code style="background-color:white">x</code> such that <code style="background-color:white">pow(x, 336 // 2, 337) != 1</code> (these are easy to find; one answer is 5), and then taking the (336 / 16)'th power of it.
</div>
</blockquote>
<p>Here’s the algorithm (in a slightly simplified form; see <a href="https://github.com/ethereum/research/blob/master/mimc_stark/fft.py">code here</a> for something slightly more optimized):</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>def fft(vals, modulus, root_of_unity):
if len(vals) == 1:
return vals
L = fft(vals[::2], modulus, pow(root_of_unity, 2, modulus))
R = fft(vals[1::2], modulus, pow(root_of_unity, 2, modulus))
o = [0 for i in vals]
for i, (x, y) in enumerate(zip(L, R)):
y_times_root = y*pow(root_of_unity, i, modulus)
o[i] = (x+y_times_root) % modulus
o[i+len(L)] = (x-y_times_root) % modulus
return o
def inv_fft(vals, modulus, root_of_unity):
f = PrimeField(modulus)
# Inverse FFT
invlen = f.inv(len(vals))
return [(x*invlen) % modulus for x in
fft(vals, modulus, f.inv(root_of_unity))]
</code></pre></div></div>
<p>You can try running it on a few inputs yourself and check that it gives results that, when you use <code class="highlighter-rouge">eval_poly_at</code> on them, give you the answers you expect to get. For example:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>>>> fft.fft([3,1,4,1,5,9,2,6], 337, 85, inv=True)
[46, 169, 29, 149, 126, 262, 140, 93]
>>> f = poly_utils.PrimeField(337)
>>> [f.eval_poly_at([46, 169, 29, 149, 126, 262, 140, 93], f.exp(85, i)) for i in range(8)]
[3, 1, 4, 1, 5, 9, 2, 6]
</code></pre></div></div>
<p>A Fourier transform takes as input <code class="highlighter-rouge">[x[0] .... x[n-1]]</code>, and its goal is to output <code class="highlighter-rouge">x[0] + x[1] + ... + x[n-1]</code> as the first element, <code class="highlighter-rouge">x[0] + x[1] * 2 + ... + x[n-1] * w**(n-1)</code> as the second element, etc etc; a fast Fourier transform accomplishes this by splitting the data in half, doing an FFT on both halves, and then gluing the result back together.</p>
<center>
<img src="https://vitalik.ca/files/radix2fft.png" /><br />
<small><i>A diagram of how information flows through the FFT computation. Notice how the FFT consists of a "gluing" step followed by two copies of the FFT on two halves of the data, and so on recursively until you're down to one element.</i></small>
</center>
<p>I recommend <a href="http://web.cecs.pdx.edu/~maier/cs584/Lectures/lect07b-11-MG.pdf">this</a> for more intuition on how or why the FFT works and polynomial math in general, and <a href="https://dsp.stackexchange.com/questions/41558/what-are-some-of-the-differences-between-dft-and-fft-that-make-fft-so-fast?rq=1">this thread</a> for some more specifics on DFT vs FFT, though be warned that most literature on Fourier transforms talks about Fourier transforms over <em>real and complex numbers</em>, not <em>prime fields</em>. If you find this too hard and don’t want to understand it, just treat it as weird spooky voodoo that just works because you ran the code a few times and verified that it works, and you’ll be fine too.</p>
<h3 id="thank-goodness-its-fri-day-thats-fast-reed-solomon-interactive-oracle-proofs-of-proximity">Thank Goodness It’s FRI-day (that’s “Fast Reed-Solomon Interactive Oracle Proofs of Proximity”)</h3>
<p><em><strong>Reminder</strong>: now may be a good time to review and re-read <a href="https://vitalik.ca/general/2017/11/22/starks_part_2.html">Part 2</a></em></p>
<p>Now, we’ll get into <a href="https://github.com/ethereum/research/blob/master/mimc_stark/fri.py">the code</a> for making a low-degree proof. To review, a low-degree proof is a (probabilistic) proof that at least some high percentage (eg. 80%) of a given set of values represent the evaluations of some specific polynomial whose degree is much lower than the number of values given. Intuitively, just think of it as a proof that “some Merkle root that we claim represents a polynomial actually does represent a polynomial, possibly with a few errors”. As input, we have:</p>
<ul>
<li>A set of values that we claim are the evaluation of a low-degree polynomial</li>
<li>A root of unity; the x coordinates at which the polynomial is evaluated are successive powers of this root of unity</li>
<li>A value N such that we are proving the degree of the polynomial is <em>strictly less than</em> N</li>
<li>The modulus</li>
</ul>
<p>Our approach is a recursive one, with two cases. First, if the degree is low enough, we just provide the entire list of values as a proof; this is the “base case”. Verification of the base case is trivial: do an FFT or Lagrange interpolation or whatever else to interpolate the polynomial representing those values, and verify that its degree is < N. Otherwise, if the degree is higher than some set minimum, we do the vertical-and-diagonal trick described <a href="https://vitalik.ca/general/2017/11/22/starks_part_2.html">at the bottom of Part 2</a>.</p>
<p>We start off by putting the values into a Merkle tree and using the Merkle root to select a pseudo-random x coordinate (<code class="highlighter-rouge">special_x</code>). We then calculate the “column”:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Calculate the set of x coordinates
xs = get_power_cycle(root_of_unity, modulus)
column = []
for i in range(len(xs)//4):
x_poly = f.lagrange_interp_4(
[xs[i+len(xs)*j//4] for j in range(4)],
[values[i+len(values)*j//4] for j in range(4)],
)
column.append(f.eval_poly_at(x_poly, special_x))
</code></pre></div></div>
<p>This packs a lot into a few lines of code. The broad idea is to re-interpret the polynomial <code class="highlighter-rouge">P(x)</code> as a polynomial <code class="highlighter-rouge">Q(x, y)</code>, where <code class="highlighter-rouge">P(x) = Q(x, x**4)</code>. If P has degree < N, then <code class="highlighter-rouge">P'(y) = Q(special_x, y)</code> will have degree < N/4. Since we don’t want to take the effort to actually compute Q in coefficient form (that would take a still-relatively-nasty-and-expensive FFT!), we instead use another trick. For any given value of x<sup>4</sup>, there are 4 corresponding values of <code class="highlighter-rouge">x</code>: <code class="highlighter-rouge">x</code>, <code class="highlighter-rouge">modulus - x</code>, and <code class="highlighter-rouge">x</code> multiplied by the two modular square roots of <code class="highlighter-rouge">-1</code>. So we already have four values of <code class="highlighter-rouge">Q(?, x**4)</code>, which we can use to interpolate the polynomial <code class="highlighter-rouge">R(x) = Q(x, x**4)</code>, and from there calculate <code class="highlighter-rouge">R(special_x) = Q(special_x, x**4) = P'(x**4)</code>. There are N/4 possible values of x<sup>4</sup>, and this lets us easily calculate all of them.</p>
<center>
<img src="https://vitalik.ca/files/fri7.png" style="width:550px" /><br />
<small><i>A diagram from part 2; it helps to keep this in mind when understanding what's going on here</i></small>
</center>
<p>Our proof consists of some number (eg. 40) of random queries from the list of values of x<sup>4</sup> (using the Merkle root of the column as a seed), and for each query we provide Merkle branches of the five values of <code class="highlighter-rouge">Q(?, x**4)</code>:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>m2 = merkelize(column)
# Pseudo-randomly select y indices to sample
# (m2[1] is the Merkle root of the column)
ys = get_pseudorandom_indices(m2[1], len(column), 40)
# Compute the Merkle branches for the values in the polynomial and the column
branches = []
for y in ys:
branches.append([mk_branch(m2, y)] +
[mk_branch(m, y + (len(xs) // 4) * j) for j in range(4)])
</code></pre></div></div>
<p>The verifier’s job will be to verify that these five values actually do lie on the same degree < 4 polynomial. From there, we recurse and do an FRI on the column, verifying that the column actually does have degree < N/4. That really is all there is to FRI.</p>
<p>As a challenge exercise, you could try creating low-degree proofs of polynomial evaluations that have errors in them, and see how many errors you can get away passing the verifier with (hint, you’ll need to modify the <code class="highlighter-rouge">prove_low_degree</code> function; with the default prover, even one error will balloon up and cause verification to fail).</p>
<h3 id="the-stark">The STARK</h3>
<p><em><strong>Reminder</strong>: now may be a good time to review and re-read <a href="https://vitalik.ca/general/2017/11/09/starks_part_1.html">Part 1</a></em></p>
<p>Now, we get to the actual meat that puts all of these pieces together: <code class="highlighter-rouge">def mk_mimc_proof(inp, steps, round_constants)</code> (code <a href="https://github.com/ethereum/research/blob/master/mimc_stark/mimc_stark.py">here</a>), which generates a proof of the execution result of running the MIMC function with the given input for some number of steps. First, some asserts:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>assert steps <= 2**32 // extension_factor
assert is_a_power_of_2(steps) and is_a_power_of_2(len(round_constants))
assert len(round_constants) < steps
</code></pre></div></div>
<p>The extension factor is the extent to which we will be “stretching” the computational trace (the set of “intermediate values” of executing the MIMC function). We need the step count multiplied by the extension factor to be at most 2<sup>32</sup>, because we don’t have roots of unity of order 2<sup>k</sup> for <code class="highlighter-rouge">k > 32</code>.</p>
<p>Our first computation will be to generate the computational trace; that is, all of the <em>intermediate</em> values of the computation, from the input going all the way to the output.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Generate the computational trace
computational_trace = [inp]
for i in range(steps-1):
computational_trace.append((computational_trace[-1]**3 + round_constants[i % len(round_constants)]) % modulus)
output = computational_trace[-1]
</code></pre></div></div>
<p>We then convert the computation trace into a polynomial, “laying down” successive values in the trace on successive powers of a root of unity <code class="highlighter-rouge">g</code> where g<sup>steps</sup> = 1, and we then evaluate the polynomial in a larger set, of successive powers of a root of unity <code class="highlighter-rouge">g2</code> where <code class="highlighter-rouge">g2</code><sup>steps * 8</sup> = 1 (note that <code class="highlighter-rouge">g2</code><sup>8</sup> = g).</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>computational_trace_polynomial = inv_fft(computational_trace, modulus, subroot)
p_evaluations = fft(computational_trace_polynomial, modulus, root_of_unity)
</code></pre></div></div>
<center>
<img src="http://vitalik.ca/files/RootsOfUnity.png" /><br />
<small><i>Black: powers of `g1`. Purple: powers of `g2`. Orange: 1. You can look at successive roots of unity as being arranged in a circle in this way. We are "laying" the computational trace along powers of `g1`, and then extending it compute the values of the same polynomial at the intermediate values (ie. the powers of `g2`).</i></small>
</center>
<p>We can convert the round constants of MIMC into a polynomial. Because these round constants loop around very frequently (in our tests, every 64 steps), it turns out that they form a degree-64 polynomial, and we can fairly easily compute its expression, and its extension:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>skips2 = steps // len(round_constants)
constants_mini_polynomial = fft(round_constants, modulus, f.exp(subroot, skips2), inv=True)
constants_polynomial = [0 if i % skips2 else constants_mini_polynomial[i//skips2] for i in range(steps)]
constants_mini_extension = fft(constants_mini_polynomial, modulus, f.exp(root_of_unity, skips2))
</code></pre></div></div>
<p>Suppose there are 8192 steps of execution and 64 round constants. Here is what we are doing: we are doing an FFT to compute the round constants <i>as a function of <code class="highlighter-rouge">g1</code><sup>128</sup></i>. We then add zeroes in between the constants to make it a function of <code class="highlighter-rouge">g1</code> itself. Because <code class="highlighter-rouge">g1</code><sup>128</sup> loops around every 64 steps, we know this function of <code class="highlighter-rouge">g1</code> will as well. We only compute 512 steps of the extension, because we know that the extension repeats after 512 steps as well.</p>
<p>We now, as in the Fibonacci example in Part 1, calculate <code class="highlighter-rouge">C(P(x))</code>, except this time it’s <code class="highlighter-rouge">C(P(x), P(g1*x), K(x))</code>:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Create the composed polynomial such that
# C(P(x), P(g1*x), K(x)) = P(g1*x) - P(x)**3 - K(x)
c_of_p_evaluations = [(p_evaluations[(i+extension_factor)%precision] -
f.exp(p_evaluations[i], 3) -
constants_mini_extension[i % len(constants_mini_extension)])
% modulus for i in range(precision)]
print('Computed C(P, K) polynomial')
</code></pre></div></div>
<p>Note that here we are no longer working with polynomials in <em>coefficient form</em>; we are working with the polynomials in terms of their evaluations at successive powers of the higher-order root of unity.</p>
<p><code class="highlighter-rouge">c_of_p</code> is intended to be <code class="highlighter-rouge">Q(x) = C(P(x), P(g1*x), K(x)) = P(g1*x) - P(x)**3 - K(x)</code>; the goal is that for every <code class="highlighter-rouge">x</code> that we are laying the computational trace along (except for the last step, as there’s no step “after” the last step), the next value in the trace is equal to the previous value in the trace cubed, plus the round constant. Unlike the Fibonacci example in Part 1, where if one computational step was at coordinate k, the next step is at coordinate k+1, here we are laying down the computational trace along successive powers of the lower-order root of unity (<code class="highlighter-rouge">g1</code>), so if one computational step is located at x = <code class="highlighter-rouge">g1</code><sup><code class="highlighter-rouge">i</code></sup>, the “next” step is located at <code class="highlighter-rouge">g1</code><sup><code class="highlighter-rouge">i+1</code></sup> = <code class="highlighter-rouge">g1</code><sup><code class="highlighter-rouge">i</code></sup> * <code class="highlighter-rouge">g1</code> = <code class="highlighter-rouge">x * g1</code>. Hence, for every power of the lower-order root of unity (<code class="highlighter-rouge">g1</code>) (except the last), we want it to be the case that <code class="highlighter-rouge">P(x*g1) = P(x)**3 + K(x)</code>, or <code class="highlighter-rouge">P(x*g1) - P(x)**3 - K(x) = Q(x) = 0</code>. Thus, <code class="highlighter-rouge">Q(x)</code> will be equal to zero at all successive powers of the lower-order root of unity g (except the last).</p>
<p>There is an algebraic theorem that proves that if <code class="highlighter-rouge">Q(x)</code> is equal to zero at all of these x coordinates, then it is a multiple of the <em>minimal</em> polynomial that is equal to zero at all of these x coordinates: <code class="highlighter-rouge">Z(x) = (x - x_1) * (x - x_2) * ... * (x - x_n)</code>. Since proving that <code class="highlighter-rouge">Q(x)</code> is equal to zero at every single coordinate we want to check is too hard (as verifying such a proof would take longer than just running the original computation!), instead we use an indirect approach to (probabilistically) prove that <code class="highlighter-rouge">Q(x)</code> is a multiple of <code class="highlighter-rouge">Z(x)</code>. And how do we do that? By providing the quotient <code class="highlighter-rouge">D(x) = Q(x) / Z(x)</code> and using FRI to prove that it’s an actual polynomial and not a fraction, of course!</p>
<p>We chose the particular arrangement of lower and higher order roots of unity (rather than, say, laying the computational trace along the first few powers of the higher order root of unity) because it turns out that computing <code class="highlighter-rouge">Z(x)</code> (the polynomial that evaluates to zero at all points along the computational trace except the last), and dividing by <code class="highlighter-rouge">Z(x)</code> is trivial there: the expression of Z is a fraction of two terms.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Compute D(x) = Q(x) / Z(x)
# Z(x) = (x^steps - 1) / (x - x_atlast_step)
z_num_evaluations = [xs[(i * steps) % precision] - 1 for i in range(precision)]
z_num_inv = f.multi_inv(z_num_evaluations)
z_den_evaluations = [xs[i] - last_step_position for i in range(precision)]
d_evaluations = [cp * zd * zni % modulus for cp, zd, zni in zip(c_of_p_evaluations, z_den_evaluations, z_num_inv)]
print('Computed D polynomial')
</code></pre></div></div>
<p>Notice that we compute the numerator and denominator of Z directly in “evaluation form”, and then use the batch modular inversion to turn dividing by Z into a multiplication (* zd * zni), and then pointwise multiply the evaluations of <code class="highlighter-rouge">Q(x)</code> by these inverses of <code class="highlighter-rouge">Z(x)</code>. Note that at the powers of the lower-order root of unity except the last (ie. along the portion of the low-degree extension that is part of the original computational trace), <code class="highlighter-rouge">Z(x) = 0</code>, so this computation involving its inverse will break. This is unfortunate, though we will plug the hole by simply modifying the random checks and FRI algorithm to not sample at those points, so the fact that we calculated them wrong will never matter.</p>
<p>Because <code class="highlighter-rouge">Z(x)</code> can be expressed so compactly, we get another benefit: the verifier can compute <code class="highlighter-rouge">Z(x)</code> for any specific <code class="highlighter-rouge">x</code> extremely quickly, without needing any precomputation. It’s okay for the <em>prover</em> to have to deal with polynomials whose size equals the number of steps, but we don’t want to ask the <em>verifier</em> to do the same, as we want verification to be succinct (ie. ultra-fast, with proofs as small as possible).</p>
<p>Probabilistically checking <code class="highlighter-rouge">D(x) * Z(x) = Q(x)</code> at a few randomly selected points allows us to verify the <strong>transition constraints</strong> - that each computational step is a valid consequence of the previous step. But we also want to verify the <strong>boundary constraints</strong> - that the input and the output of the computation is what the prover says they are. Just asking the prover to provide evaluations of <code class="highlighter-rouge">P(1)</code>, <code class="highlighter-rouge">D(1)</code>, <code class="highlighter-rouge">P(last_step)</code> and <code class="highlighter-rouge">D(last_step)</code> (where <code class="highlighter-rouge">last_step</code> (or g<sup>steps-1</sup>) is the coordinate corresponding to the last step in the computation) is too fragile; there’s no proof that those values are on the same polynomial as the rest of the data. So instead we use a similar kind of polynomial division trick:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Compute interpolant of ((1, input), (x_atlast_step, output))
interpolant = f.lagrange_interp_2([1, last_step_position], [inp, output])
i_evaluations = [f.eval_poly_at(interpolant, x) for x in xs]
zeropoly2 = f.mul_polys([-1, 1], [-last_step_position, 1])
inv_z2_evaluations = f.multi_inv([f.eval_poly_at(quotient, x) for x in xs])
# B = (P - I) / Z2
b_evaluations = [((p - i) * invq) % modulus for p, i, invq in zip(p_evaluations, i_evaluations, inv_z2_evaluations)]
print('Computed B polynomial')
</code></pre></div></div>
<p>The argument is as follows. The prover wants to prove <code class="highlighter-rouge">P(1) == input</code> and <code class="highlighter-rouge">P(last_step) == output</code>. If we take <code class="highlighter-rouge">I(x)</code> as the <em>interpolant</em> - the line that crosses the two points <code class="highlighter-rouge">(1, input)</code> and <code class="highlighter-rouge">(last_step, output)</code>, then <code class="highlighter-rouge">P(x) - I(x)</code> would be equal to zero at those two points. Thus, it suffices to prove that <code class="highlighter-rouge">P(x) - I(x)</code> is a multiple of <code class="highlighter-rouge">(x - 1) * (x - last_step)</code>, and we do that by… providing the quotient!</p>
<center>
<img src="http://vitalik.ca/files/P_I_and_B.png" /><img src="http://vitalik.ca/files/P_I_and_B_2.png" /><br />
<small><i>Purple: computational trace polynomial (P). Green: interpolant (I) (notice how the interpolant is constructed to equal the input (which should be the first step of the computational trace) at x=1 and the output (which should be the last step of the computational trace) at x=g<sup>steps-1</sup>. Red: P - I. Yellow: the minimal polynomial that equals 0 at x=1 and x=g<sup>steps-1</sup> (that is, Z2). Pink: (P - I) / Z2.</i></small>
</center>
<p><br /></p>
<blockquote><b>Challenge</b>
Suppose you wanted to <i>also</i> prove that the value in the computational trace after the 703rd computational step is equal to 8018284612598740. How would you modify the above algorithm to do that?
<br />
<b>Mouseover below for answer</b>
<br />
<div class="foo">
Set <code style="background-color:white">I(x)</code> to be the interpolant of <code style="background-color:white">(1, input), (g ** 703, 8018284612598740), (last_step, output)</code>, and make a proof by providing the quotient <code style="background-color:white">B(x) = (P(x) - I(x)) / ((x - 1) * (x - g ** 703) * (x - last_step))</code>
<br />
</div>
</blockquote>
<p>Now, we commit to the Merkle root of P, D and B combined together.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Compute their Merkle roots
mtree = merkelize([pval.to_bytes(32, 'big') +
dval.to_bytes(32, 'big') +
bval.to_bytes(32, 'big') for
pval, dval, bval in zip(p_evaluations, d_evaluations, b_evaluations)])
print('Computed hash root')
</code></pre></div></div>
<p>Now, we need to prove that P, D and B are all actually polynomials, and of the right max-degree. But FRI proofs are big and expensive, and we don’t want to have three FRI proofs. So instead, we compute a pseudorandom linear combination of P, D and B (using the Merkle root of P, D and B as a seed), and do an FRI proof on that:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>k1 = int.from_bytes(blake(mtree[1] + b'\x01'), 'big')
k2 = int.from_bytes(blake(mtree[1] + b'\x02'), 'big')
k3 = int.from_bytes(blake(mtree[1] + b'\x03'), 'big')
k4 = int.from_bytes(blake(mtree[1] + b'\x04'), 'big')
# Compute the linear combination. We don't even bother calculating it
# in coefficient form; we just compute the evaluations
root_of_unity_to_the_steps = f.exp(root_of_unity, steps)
powers = [1]
for i in range(1, precision):
powers.append(powers[-1] * root_of_unity_to_the_steps % modulus)
l_evaluations = [(d_evaluations[i] +
p_evaluations[i] * k1 + p_evaluations[i] * k2 * powers[i] +
b_evaluations[i] * k3 + b_evaluations[i] * powers[i] * k4) % modulus
for i in range(precision)]
</code></pre></div></div>
<p>Unless all three of the polynomials have the right low degree, it’s almost impossible that a randomly selected linear combination of them will (you have to get <em>extremely</em> lucky for the terms to cancel), so this is sufficient.</p>
<p>We want to prove that the degree of D is less than <code class="highlighter-rouge">2 * steps</code>, and that of P and B are less than <code class="highlighter-rouge">steps</code>, so we actually make a random linear combination of P, P * x<sup>steps</sup>, B, B<sup>steps</sup> and D, and check that the degree of this combination is less than <code class="highlighter-rouge">2 * steps</code>.</p>
<p>Now, we do some spot checks of all of the polynomials. We generate some random indices, and provide the Merkle branches of the polynomial evaluated at those indices:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Do some spot checks of the Merkle tree at pseudo-random coordinates, excluding
# multiples of `extension_factor`
branches = []
samples = spot_check_security_factor
positions = get_pseudorandom_indices(l_mtree[1], precision, samples,
exclude_multiples_of=extension_factor)
for pos in positions:
branches.append(mk_branch(mtree, pos))
branches.append(mk_branch(mtree, (pos + skips) % precision))
branches.append(mk_branch(l_mtree, pos))
print('Computed %d spot checks' % samples)
</code></pre></div></div>
<p>The <code class="highlighter-rouge">get_pseudorandom_indices</code> function returns some random indices in the range [0…precision-1], and the <code class="highlighter-rouge">exclude_multiples_of</code> parameter tells it to not give values that are multiples of the given parameter (here, <code class="highlighter-rouge">extension_factor</code>). This ensures that we do not sample along the original computational trace, where we are likely to get wrong answers.</p>
<p>The proof (~250-500 kilobytes altogether) consists of a set of Merkle roots, the spot-checked branches, and a low-degree proof of the random linear combination:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>o = [mtree[1],
l_mtree[1],
branches,
prove_low_degree(l_evaluations, root_of_unity, steps * 2, modulus, exclude_multiples_of=extension_factor)]
</code></pre></div></div>
<p>The largest parts of the proof in practice are the Merkle branches, and the FRI proof, which consists of even more branches. And here’s the “meat” of the verifier:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>for i, pos in enumerate(positions):
x = f.exp(G2, pos)
x_to_the_steps = f.exp(x, steps)
mbranch1 = verify_branch(m_root, pos, branches[i*3])
mbranch2 = verify_branch(m_root, (pos+skips)%precision, branches[i*3+1])
l_of_x = verify_branch(l_root, pos, branches[i*3 + 2], output_as_int=True)
p_of_x = int.from_bytes(mbranch1[:32], 'big')
p_of_g1x = int.from_bytes(mbranch2[:32], 'big')
d_of_x = int.from_bytes(mbranch1[32:64], 'big')
b_of_x = int.from_bytes(mbranch1[64:], 'big')
zvalue = f.div(f.exp(x, steps) - 1,
x - last_step_position)
k_of_x = f.eval_poly_at(constants_mini_polynomial, f.exp(x, skips2))
# Check transition constraints Q(x) = Z(x) * D(x)
assert (p_of_g1x - p_of_x ** 3 - k_of_x - zvalue * d_of_x) % modulus == 0
# Check boundary constraints B(x) * Z2(x) + I(x) = P(x)
interpolant = f.lagrange_interp_2([1, last_step_position], [inp, output])
zeropoly2 = f.mul_polys([-1, 1], [-last_step_position, 1])
assert (p_of_x - b_of_x * f.eval_poly_at(zeropoly2, x) -
f.eval_poly_at(interpolant, x)) % modulus == 0
# Check correctness of the linear combination
assert (l_of_x - d_of_x -
k1 * p_of_x - k2 * p_of_x * x_to_the_steps -
k3 * b_of_x - k4 * b_of_x * x_to_the_steps) % modulus == 0
</code></pre></div></div>
<p>At every one of the positions that the prover provides a Merkle proof for, the verifier checks the Merkle proof, and checks that <code class="highlighter-rouge">C(P(x), P(g1*x), K(x)) = Z(x) * D(x)</code> and <code class="highlighter-rouge">B(x) * Z2(x) + I(x) = P(x)</code> (reminder: for <code class="highlighter-rouge">x</code> that are not along the original computation trace, <code class="highlighter-rouge">Z(x)</code> will not be zero, and so <code class="highlighter-rouge">C(P(x), P(g1*x), K(x))</code> likely will not evaluate to zero). The verifier also checks that the linear combination is correct, and calls <code class="highlighter-rouge">verify_low_degree_proof(l_root, root_of_unity, fri_proof, steps * 2, modulus, exclude_multiples_of=extension_factor)</code> to verify the FRI proof. <strong>And we’re done</strong>!</p>
<p>Well, not really; soundness analysis to prove how many spot-checks for the cross-polynomial checking and for the FRI are necessary is really tricky. But that’s all there is to the code, at least if you don’t care about making even crazier optimizations. When I run the code above, we get a STARK proving “overhead” of about 300-400x (eg. a MIMC computation that takes 0.2 seconds to calculate takes 60 second to prove), suggesting that with a 4-core machine computing the STARK of the MIMC computation in the forward direction could actually be faster than computing MIMC in the backward direction. That said, these are both relatively inefficient implementations in python, and the proving to running time ratio for properly optimized implementations may be different. Also, it’s worth pointing out that the STARK proving overhead for MIMC is remarkably low, because MIMC is almost perfectly “arithmetizable” - it’s mathematical form is very simple. For “average” computations, which contain less arithmetically clean operations (eg. checking if a number is greater or less than another number), the overhead is likely much higher, possibly around 10000-50000x.</p>
Sat, 21 Jul 2018 18:03:10 -0700
https://vitalik.ca/general/2018/07/21/starks_part_3.html
https://vitalik.ca/general/2018/07/21/starks_part_3.htmlgeneralOn Radical Markets<p>Recently I had the fortune to have received an advance copy of Eric Posner and Glen Weyl’s new book, <em><a href="https://www.amazon.ca/dp/B0773X7RKB/ref=dp-kindle-redirect?_encoding=UTF8&btkr=1">Radical Markets</a></em>, which could be best described as an interesting new way of looking at the subject that is sometimes called “<a href="https://en.wikipedia.org/wiki/Political_economy">political economy</a>” - tackling the big questions of how markets and politics and society intersect. The general philosophy of the book, as I interpret it, can be expressed as follows. Markets are great, and price mechanisms are an awesome way of guiding the use of resources in society and bringing together many participants’ objectives and information into a coherent whole. However, markets are socially constructed because they depend on property rights that are socially constructed, and there are many different ways that markets and property rights can be constructed, some of which are unexplored and potentially far better than what we have today. Contra doctrinaire libertarians, freedom is a high-dimensional design space.</p>
<p>The book interests me for multiple reasons. First, although I spend most of my time in the blockchain/crypto space heading up the Ethereum project and in some cases providing various kinds of support to projects in the space, I do also have broader interests, of which the use of economics and mechanism design to make more open, free, egalitarian and efficient systems for human cooperation, including improving or replacing present-day corporations and governments, is a major one. The intersection of interests between the Ethereum community and Posner and Weyl’s work is multifaceted and plentiful; <em>Radical Markets</em> dedicates an entire chapter to the idea of “markets for personal data”, redefining the economic relationship between ourselves and services like Facebook, and well, look what the Ethereum community is working on: <a href="https://cointelegraph.com/news/blockchain-startup-can-help-consumers-profit-from-their-personal-data">markets</a> <a href="https://cointelegraph.com/news/marketplace-aims-to-resell-personal-data-and-create-passive-income-stream-for-users">for</a> <a href="https://datum.org/">personal</a> <a href="https://blog.enigma.co/the-enigma-data-marketplace-is-live-84a269ec17fb">data</a>.</p>
<p>Second, blockchains may well be used as a technical backbone for some of the solutions described in the book, and Ethereum-style smart contracts are ideal for the kinds of complex systems of property rights that the book explores. Third, the economic ideas and challenges that the book brings up are ideas that have also been explored, and will be continue to be explored, at great length by the blockchain community for its own purposes. Posner and Weyl’s ideas often have the feature that they allow economic incentive alignment to serve as a substitute for subjective ad-hoc bureaucracy (eg. Harberger taxes can essentially replace <a href="https://en.wikipedia.org/wiki/Eminent_domain">eminent domain</a>), and given that blockchains lack access to trusted human-controlled courts, these kinds of solutions may prove to be be even more ideal for blockchain-based markets than they are for “real life”.</p>
<p>I will warn that readers are not at all guaranteed to find the book’s proposals acceptable; at least the first three have <a href="https://www.politico.com/magazine/story/2018/02/13/immigration-visas-economics-216968">already been</a> highly controversial and they do contravene many people’s moral preconceptions about how property should and should work and where money and markets can and can’t be used. The authors are no strangers to controversy; Posner has on previous occasions even <a href="https://www.theguardian.com/news/2014/dec/04/-sp-case-against-human-rights">proven willing</a> to argue against such notions as human rights law. That said, the book does go to considerable lengths to explain why each proposal improves efficiency if it could be done, and offer multiple versions of each proposal in the hopes that there is at least one (even if partial) implementation of each idea that any given reader can find agreeable.</p>
<h2 id="what-do-posner-and-weyl-talk-about">What do Posner and Weyl talk about?</h2>
<p>The book is split into five major sections, each arguing for a particular reform: self-assessed property taxes, quadratic voting, a new kind of immigration program, breaking up big financial conglomerates that currently make banks and other industries act like monopolies even if they appear at first glance to be competitive, and markets for selling personal data. Properly summarizing all five sections and doing them justice would take too long, so I will focus on a deep summary of one specific section, dealing with a new kind of property taxation, to give the reader a feel for the kinds of ideas that the book is about.</p>
<h3 id="harberger-taxes">Harberger taxes</h3>
<p><em>See also: “<a href="https://chicagounbound.uchicago.edu/cgi/viewcontent.cgi?article=12668&context=journal_articles">Property Is Only Another Name for Monopoly</a>”, Posner and Weyl</em></p>
<p>Markets and private property are two ideas that are often considered together, and it is difficult in modern discourse to imagine one without (or even with much less of) the other. In the 19th century, however, many economists in Europe were both libertarian <em>and</em> egalitarian, and it was quite common to appreciate markets while maintaining skepticism toward the excesses of private property. A rather interesting example of this is the <a href="http://praxeology.net/FB-PJP-DOI.htm">Bastiat-Proudhon debate</a> from 1849-1850 where the two dispute the legitimacy of charging interest on loans, with one side focusing on the mutual gains from voluntary contracts and the other focusing on their suspicion of the potential for people with capital to get even richer without working, leading to unbalanced capital accumulation.</p>
<p>As it turns out, it is absolutely possible to have a system that contains markets but not property rights: at the end of every year, collect every piece of property, and at the start of the next year have the government auction every piece out to the highest bidder. This kind of system is intuitively quite unrealistic and impractical, but it has the benefit that it achieves perfect <strong>allocative efficiency</strong>: every year, every object goes to the person who can derive the most value from it (ie. the highest bidder). It also gives the government a large amount of revenue that could be used to completely substitute income and sales taxes or fund a basic income.</p>
<p>Now you might ask: doesn’t the existing property system also achieve allocative efficiency? After all, if I have an apple, and I value it at $2, and you value it at $3, then you could offer me $2.50 and I would accept. However, this fails to take into account imperfect information: how do you know that I value it at $2, and not $2.70? You could offer to buy it for $2.99 so that you can be sure that you’ll get it if you really are the one who values the apple more, but then you would be gaining practically nothing from the transaction. And if you ask me to set the price, how do I know that you value it at $3, and not $2.30? And if I set the price to $2.01 to be sure, I would be gaining practically nothing from the transaction. Unfortunately, there is a result known as the <a href="https://en.wikipedia.org/wiki/Myerson%E2%80%93Satterthwaite_theorem">Myerson-Satterthwaite Theorem</a> which means that <em>no</em> solution is efficient; that is, any bargaining algorithm in such a situation must at least sometimes lead to inefficiency from mutually beneficial deals falling through.</p>
<p>If there are many buyers you have to negotiate with, things get even harder. If a developer (in the real estate sense) is trying to make a large project that requires buying 100 existing properties, and 99 have already agreed, the remaining one has a strong incentive to charge a very high price, much higher than their actual personal valuation of the property, hoping that the developer will have no choice but to pay up.</p>
<center>
<img src="https://nationalpostcom.files.wordpress.com/2012/11/highway-built-around-house03.jpg" style="width:450px" /><br />
<small><i>Well, not necessarily no choice. But a very inconvenient and both privately and socially wasteful choice.</i></small>
</center>
<p><br /></p>
<p>Re-auctioning everything once a year completely solves this problem of allocative efficiency, but at a very high cost to <strong>investment efficiency</strong>: there’s no point in building a house in the first place if six months later it will get taken away from you and re-sold in an auction. All property taxes have this problem; if building a house costs you $90 and brings you $100 of benefit, but then you have to pay $15 more property tax if you build the house, then you will not build the house and that $10 gain is lost to society.</p>
<p>One of the more interesting ideas from the 19th century economists, and specifically Henry George, was a kind of property tax that did not have this problem: the <a href="https://en.wikipedia.org/wiki/Land_value_tax">land value tax</a>. The idea is to charge tax on the value of land, but not the <em>improvements to the land</em>; if you own a $100,000 plot of dirt you would have to pay $5,000 per year taxes on it regardless of whether you used the land to build a condominium or simply as a place to walk your pet doge.</p>
<center>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/58/Shiba_inu_taiki.jpg/220px-Shiba_inu_taiki.jpg" style="width:250px" /><br />
<small><i>A doge.</i></small>
</center>
<p><br /></p>
<p>Weyl and Posner are not convinced that Georgian land taxes are viable in practice:</p>
<blockquote>
<p>Consider, for example, the Empire State Building. What is the pure value of the land beneath it? One could try to infer its value by comparing it to the value of adjoining land. But the building itself defines the neighborhood around it; removing the building would almost certainly change the value of its surrounding land. The land and the building, even the neighborhood, are so tied together, it would be hard to figure out a separate value for each of them.</p>
</blockquote>
<p>Arguably this does not exclude the possibility of a different kind of Georgian-style land tax: a tax based on the <em>average</em> of property values across a sufficiently large area. That would preserve the property that improving a single piece of land would not (greatly) perversely increase the taxes that they have to pay, without having to find a way to distinguish land from improvements in an absolute sense. But in any case, Posner and Weyl move on to their main proposal: self-assessed property taxes.</p>
<p>Consider a system where property owners themselves specify what the value of their property is, and pay a tax rate of, say, 2% of that value per year. But here is the twist: whatever value they specify for their property, <em>they have to be willing to sell it to anyone at that price</em>.</p>
<p>If the tax rate is equal to the chance per year that the property gets sold, then this achieves optimal allocative efficiency: raising your self-assessed property value by $1 increases the tax you pay by $0.02, but it also means there is a 2% chance that someone will buy the property and pay $1 more, so there is no incentive to cheat in either direction. It does harm investment efficiency, but vastly less so than all property being re-auctioned every year.</p>
<p>Posner and Weyl then point out that if more investment efficiency is desired, a hybrid solution with a lower property tax is possible:</p>
<blockquote>
<p>When the tax is reduced incrementally to improve investment efficiency, the loss in allocative efficiency is less than the gain in investment efficiency. The reason is that the most valuable sales are ones where the buyer is willing to pay significantly more than the seller is willing to accept. These transactions are the first ones enabled by a reduction in the price as even a small price reduction will avoid blocking these most valuable transactions. In fact, it can be shown that the size of the social loss from monopoly power grows quadratically in the extent of this power. Thus, reducing the markup by a third eliminates close to 5/9 = (3<sup>2</sup>-2<sup>2</sup>)/(3<sup>2</sup>) of the allocative harm from private ownership.</p>
</blockquote>
<p>This concept of quadratic deadweight loss is a truly important insight in economics, and is arguably the deep reason why “moderation in all things” is such an attractive principle: the first step you take away from an extreme will generally be the most valuable.</p>
<center>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Deadweight-loss-price-ceiling.svg/350px-Deadweight-loss-price-ceiling.svg.png" style="width:300px" />
</center>
<p><br /></p>
<p>The book then proceeds to give a series of side benefits that this tax would have, as well as some downsides. One interesting side benefit is that it removes an information asymmetry flaw that exists with property sales today, where owners have the incentive to expend effort on making their property look good even in potentially misleading ways. With a properly set Harberger tax, if you somehow mange to trick the world into thinking your house is 5% more valuable, you’ll get 5% more when you sell it but until that point you’ll have to pay 5% more in taxes, or else someone will much more quickly snap it up from you at the original price.</p>
<p>The downsides are smaller than they seem; for example, one natural disadvantage is that it exposes property owners to uncertainty due to the possibility that someone will snap up their property at any time, but that is hardly an unknown as it’s a risk that renters already face every day. But Weyl and Posner <em>do</em> propose more moderate ways of introducing the tax that don’t have these issues. First, the tax can be applied to types of property that are currently government owned; it’s a potentially superior alternative to both continued government ownership <em>and</em> traditional full-on privatization. Second, the tax can be applied to forms of property that are already “industrial” in usage: radio spectrum licenses, domain names, intellectual property, etc.</p>
<h3 id="the-rest-of-the-book">The Rest of the Book</h3>
<p>The remaining chapters bring up similar ideas that are similar in spirit to the discussion on Harberger taxes in their use of modern game-theoretic principles to make mathematically optimized versions of existing social institutions. One of the proposals is for something called quadratic voting, which I summarize as follows.</p>
<p>Suppose that you can vote as many times as you want, but voting costs “voting tokens” (say each citizen is assigned N voting tokens per year), and it costs tokens in a nonlinear way: your first vote costs one token, your second vote costs two tokens, and so forth. If someone feels more strongly about something, the argument goes, they would be willing to pay more for a single vote; quadratic voting takes advantage of this by perfectly aligning <em>quantity</em> of votes with <em>cost</em> of votes: if you’re willing to pay up to 15 tokens for a vote, then you will keep buying votes until your last one costs 15 tokens, and so you will cast 15 votes in total. If you’re willing to pay up to 30 tokens for a vote, then you will keep buying votes until you can’t buy any more for a price less than or equal to 30 tokens, and so you will end up casting 30 votes. The voting is “quadratic” because the total amount you pay for N votes goes up proportionately to N<sup>2</sup>.</p>
<center>
<img src="http://vitalik.ca/files/quadratic_voting.png" style="width:300px" />
</center>
<p><br /></p>
<p>After this, the book describes a market for immigration visas that could greatly expand the number of immigrants admitted while making sure local residents benefit and at the same time aligning incentives to encourage visa sponsors to choose immigrants that are more ikely to succeed in the country and less likely to commit crimes, then an enhancement to antitrust law, and finally the idea of setting up markets for personal data.</p>
<h3 id="markets-in-everything">Markets in Everything</h3>
<p>There are plenty of ways that one could respond to each individual proposal made in the book. I personally, for example, find the immigration visa scheme that Posner and Weyl propose well-intentioned and see how it could improve on the status quo, but also overcomplicated, and it seems simpler to me to have a scheme where visas are auctioned or sold every year, with an additional requirement for migrants to obtain liability insurance. Robin Hanson recently <a href="https://www.overcomingbias.com/2018/01/privately-enforced-punished-crime.html">proposed</a> greatly expanding liability insurance mandates as an alternative to many kinds of regulation, and while imposing new mandates on an entire society seems unrealistic, a new expanded immigration program seems like the perfect place to start considering them. Paying people for personal data is interesting, but there are concerns about adverse selection: to put it politely, the kinds of people that are willing to sit around submitting lots of data to Facebook all year to earn $16.92 (Facebook’s current <a href="https://www.cnbc.com/2017/05/03/facebook-average-revenue-per-user-arpu-q1-2017.html">annualized revenue per user</a>) are <em>not</em> the kinds of people that advertisers are willing to burn hundreds of dollars per person trying to market rolexes and Lambos to. However, what I find more interesting is the general principle that the book tries to promote.</p>
<p>Over the last hundred years, there truly has been a large amount of research into designing economic mechanisms that have desirable properties and that outperform simple two-sided buy-and-sell markets. Some of this research has been put into use in some specific industries; for example, <a href="https://en.wikipedia.org/wiki/Combinatorial_auction">combinatorial auctions</a> are used in airports, radio spectrum auctions and several other industrial use cases, but it hasn’t really seeped into any kind of broader policy design; the political systems and property rights that we have are still largely the same as we had two centuries ago. So can we use modern economic insights to reform base-layer markets and politics in such a deep way, and if so, should we?</p>
<p>Normally, I love markets and clean incentive alignment, and dislike politics and bureaucrats and ugly hacks, and I love economics, and I so love the idea of using economic insights to design markets that work better so that we can reduce the role of politics and bureaucrats and ugly hacks in society. Hence, naturally, I love this vision. So let me be a good intellectual citizen and do my best to try to make a case against it.</p>
<p>There is a limit to how complex economic incentive structures and markets can be because there is a limit to users’ ability to think and re-evaluate and give ongoing precise measurements for their valuations of things, and people value reliability and certainty. Quoting <a href="http://www.interfluidity.com/v2/5822.html">Steve Waldman criticizing Uber surge pricing</a>:</p>
<blockquote>
<p>Finally, we need to consider questions of economic calculation. In macroeconomics, we sometimes face tradeoffs between an increasing and unpredictably variable price-level and full employment. Wisely or not, our current policy is to stabilize the price level, even at short-term cost to output and employment, because stable prices enable longer-term economic calculation. That vague good, not visible on a supply/demand diagram, is deemed worth very large sacrifices. The same concern exists in a microeconomic context. If the “ride-sharing revolution” really takes hold, a lot of us will have decisions to make about whether to own a car or rely upon the Sidecars, Lyfts, and Ubers of the world to take us to work every day. To make those calculations, we will need something like predictable pricing. Commuting to our minimum wage jobs (average is over!) by Uber may be OK at standard pricing, but not so OK on a surge. In the desperate utopia of the “free-market economist”, there is always a solution to this problem. We can define futures markets on Uber trips, and so hedge our exposure to price volatility! In practice that is not so likely…</p>
</blockquote>
<p>And:</p>
<blockquote>
<p>It’s clear that in a lot of contexts, people have a strong preference for price-predictability over immediate access. The vast majority of services that we purchase and consume are not price-rationed in any fine-grained way. If your hairdresser or auto mechanic is busy, you get penciled in for next week…</p>
</blockquote>
<p>Strong property rights are valuable for the same reason: beyond the arguments about allocative and investment efficiency, they provide the mental convenience and planning benefits of predictability.</p>
<p>It’s worth noting that even Uber itself doesn’t do surge pricing in the “market-based” way that economists would recommend. Uber is not a market where drivers can set their own prices, riders can see what prices are available, and themselves choose their tradeoff between price and waiting time. Why does Uber not do this? One argument is that, as Steve Waldman says, “Uber itself is a cartel”, and wants to have the power to adjust market prices not just for efficiency but also reasons such as profit maximization, strategically setting prices to drive out competing platforms (and taxis and public transit), and public relations. As Waldman further points out, one Uber competitor, Sidecar, <em>does</em> have the ability for <a href="https://www.side.cr/drivers/">drivers to set prices</a>, and I would add that I have seen ride-sharing apps in China where <em>passengers</em> can offer drivers higher prices to try to coax them to get a car faster.</p>
<p>A possible counter-argument that Uber might give is that drivers themselves are actually less good at setting optimal prices than Uber’s own algorithms, and in general people value the convenience of one-click interfaces over the mental complexity of thinking about prices. If we assume that Uber won its market dominance over competitors like Sidecar fairly, then the market itself has decided that the economic gain from marketizing more things is not worth the mental transaction costs.</p>
<p>Harberger taxes, at least to me, seem like they would lead to these exact kinds of issues multipled by ten; people are not experts at property valuation, and would have to spend a significant amount of time and mental effort figuring out what self-assessed value to put for their house, and they would complain much more if they accidentally put a value that’s too low and suddenly find that their house is gone. If Harberger taxes were to be applied to smaller property items as well, people would need to juggle a large amount of mental valuations of everything. A similar critique could apply to many kinds of personal data markets, and possibly even to quadratic voting if implemented in its full form.</p>
<p>I could challenge this by saying “ah, even if that’s true, this is the 21st century, we could have companies that build AIs that make pricing decisions on your behalf, and people could choose the AI that seems to work best; there could even be a public option”; and Posner and Weyl themselves suggest that this is likely the way to go. And this is where the interesting conversation starts.</p>
<h3 id="tales-from-crypto-land">Tales from Crypto Land</h3>
<p>One reason why this discussion particularly interests me is that the cryptocurrency and blockchain space itself has, in some cases, run up against similar challenges. In the case of Harberger taxes, we actually did consider almost exactly that same proposal in the context of the <a href="https://ens.domains/">Ethereum Name System</a> (our decentralized alternative to DNS), but the proposal was ultimately rejected. I asked the ENS developers why it was rejected. Paraphrasing their reply, the challenge is as follows.</p>
<p>Many ENS domain names are of a type that would only be interesting to precisely two classes of actors: (i) the “legitimate owner” of some given name, and (ii) scammers. Furthermore, in some particular cases, the legitimate owner is uniquely underfunded, and scammers are uniquely dangerous. One particular case is <a href="http://myetherwallet.com">MyEtherWallet</a>, an Ethereum wallet provider. MyEtherWallet provides an important public good to the Ethereum ecosystem, making Ethereum easier to use for many thousands of people, but is able to capture only a very small portion of the value that it provides; as a result, the budget that it has for outbidding others for the domain name is low. If a scammer gets their hands on the domain, users trusting MyEtherWallet could easily be tricked into sending all of their ether (or other Ethereum assets) to a scammer. Hence, because there is generally one clear “legitimate owner” for any domain name, a pure property rights regime presents little allocative efficiency loss, and there is a strong overriding public interest toward stability of reference (ie. a domain that’s legitimate one day doesn’t redirect to a scam the next day), so <em>any</em> level of Harberger taxation may well bring more harm than good.</p>
<p>I suggested to the ENS developers the idea of applying Harberger taxes to short domains (eg. abc.eth), but not long ones; the reply was that it would be too complicated to have two classes of names. That said, perhaps there is some version of the proposal that could satisfy the specific constraints here; I would be interested to hear Posner and Weyl’s feedback on this particular application.</p>
<p>Another story from the blockchain and Ethereum space that has a more pro-radical-market conclusion is that of transaction fees. The notion of <a href="http://nakamotoinstitute.org/literature/micropayments-and-mental-transaction-costs/">mental transaction costs</a>, the idea that the inconvenience of even thinking about whether or not some small payment for a given digital good is worth it is enough of a burden to prevent “micro-markets” from working, is often used as an argument for why mass adoption of blockchain tech would be difficult: every transaction requires a small fee, and the mental expenditure of figuring out what fee to pay is itself a major usability barrier. These arguments increased further at the end of last year, when both <a href="https://bitinfocharts.com/comparison/bitcoin-transactionfees.html">Bitcoin</a> and <a href="https://bitinfocharts.com/comparison/ethereum-transactionfees.html">Ethereum</a> transaction fees briefly spiked up by a factor of over 100 due to high usage (talk about surge pricing!), and those who accidentally did not pay high enough fees saw their transactions get stuck for days.</p>
<p>That said, this is a problem that we have now, arguably, to a large extent overcome. After the spikes at the end of last year, Ethereum wallets developed more advanced algorithms for choosing what transaction fees to pay to ensure that one’s transaction gets included in the chain, and today most users are happy to simply defer to them. In my own personal experience, the mental transaction costs of worrying about transaction fees do not really exist, much like a driver of a car does not worry about the gasoline consumed by every single turn, acceleration and braking made by their car.</p>
<center>
<img src="http://vitalik.ca/files/metamask1.png" style="width:200px" /><br />
<small><i>Personal price-setting AIs for interacting with open markets: already a reality in the Ethereum transaction fee market</i></small>
</center>
<p><br /></p>
<p>A third kind of “radical market” that we are considering implementing in the context of Ethereum’s consensus system is one for incentivizing deconcentration of validator nodes in <a href="https://medium.com/@jonchoi/ethereum-casper-101-7a851a4f1eb0">proof of stake consensus</a>. It’s important for blockchains to be decentralized, a similar challenge to what antitrust law tries to solve, but the tools at our disposal are different. Posner and Weyl’s solution to antitrust, banning institutional investment funds from owning shares in multiple competitors in the same industry, is far too subjective and human-judgement-dependent to work in a blockchain, but for our specific context we have a different solution: if a validator node commits an error, it gets penalized an amount proportional to the number of other nodes that have committed an error around the same time. This incentivizes nodes to set themselves up in such a way that their failure rate is maximally uncorrelated with everyone else’s failure rate, reducing the chance that many nodes fail at the same time and threaten to the blockchain’s integrity. I want to ask Posner and Weyl: though our exact approach is fairly application-specific, could a similarly elegant “market-based” solution be discovered to incentivize market deconcentration in general?</p>
<p>All in all, I am optimistic that the various behavioral kinks around implementing “radical markets” in practice could be worked out with the help of good defaults and personal AIs, though I do think that if this vision is to be pushed forward, the greatest challenge will be finding progressively larger and more meaningful places to test it out and show that the model works. I particularly welcome the use of the blockchain and crypto space as a testing ground.</p>
<h3 id="another-kind-of-radical-market">Another Kind of Radical Market</h3>
<p>The book as a whole tends to focus on centralized reforms that could be implemented on an economy from the top down, even if their intended long-term effect is to push more decision-making power to individuals. The proposals involve large-scale restructurings of how property rights work, how voting works, how immigration and antitrust law works, and how individuals see their relationship with property, money, prices and society. But there is also the potential to use economics and game theory to come up with <em>decentralized</em> economic institutions that could be adopted by smaller groups of people at a time.</p>
<p>Perhaps the most famous examples of decentralized institutions from game theory and economics land are (i) assurance contracts, and (ii) prediction markets. An assurance contract is a system where some public good is funded by giving anyone the opportunity to pledge money, and only collecting the pledges if the total amount pledged exceeds some threshold. This ensures that people can donate money knowing that either they will get their money back or there actually will be enough to achieve some objective. A possible extension of this concept is Alex Tabarrok’s <a href="https://en.wikipedia.org/wiki/Assurance_contract#Dominant_assurance_contracts">dominant assurance contracts</a>, where an entrepreneur offers to refund participants <em>more</em> than 100% of their deposits if a given assurance contract does not raise enough money.</p>
<p>Prediction markets allow people to bet on the probability that events will happen, potentially even conditional on some action being taken (“I bet $20 that unemployment will go down if candidate X wins the election”); there are techniques for people interested in the information to subsidize the markets. Any attempt to manipulate the probability that a prediction market shows simply creates an opportunity for people to earn free money (yes I know, risk aversion and capital efficiency etc etc; still close to free) by betting against the manipulator.</p>
<p>Posner and Weyl do give one example of what I would call a decentralized institution: a game for choosing who gets an asset in the event of a divorce or a company splitting in half, where both sides provide their own valuation, the person with the higher valuation gets the item, but they must then give an amount equal to half the average of the two valuations to the loser. There’s some economic reasoning by which this solution, while not perfect, is still close to mathematically optimal.</p>
<p>One particular category of decentralized institutions I’ve been interested in is improving incentivization for content posting and content curation in social media. Some ideas that I have had include:</p>
<ul>
<li><a href="https://ethresear.ch/t/conditional-proof-of-stake-hashcash/1301">Proof of stake conditional hashcash</a> (when you send someone an email, you give them the opportunity to burn $0.5 of your money if they think it’s spam)</li>
<li><a href="https://ethresear.ch/t/prediction-markets-for-content-curation-daos/1312">Prediction markets for content curation</a> (use prediction markets to predict the results of a moderation vote on content, thereby encouraging a market of fast content pre-moderators while penalizing manipulative pre-moderation)</li>
<li>Conditional payments for paywalled content (after you pay for a piece of downloadable content and view it, you can decide after the fact if payments should go to the author or to proportionately refund previous readers)</li>
</ul>
<p>And ideas I have had in other contexts:</p>
<ul>
<li><a href="https://ethresear.ch/t/call-out-assurance-contracts/466">Call-out assurance contracts</a></li>
<li><a href="https://ethresear.ch/t/explanation-of-daicos/465">DAICOs</a> (a more decentralized and safer alternative to ICOs)</li>
</ul>
<center>
<img src="https://cryptobriefing.com/wp-content/uploads/2018/02/Buterin-Copycat-Poster-1.png" style="width:400px" /><br />
<small><i>Twitter scammers: can prediction markets incentivize an autonomous swarm of human and AI-driven moderators to flag these posts and warn users not to send them ether within a few seconds of the post being made? And could such a system be generalized to the entire internet, where these is no single centralized moderator that can easily take posts down?</i></small>
</center>
<p><br /></p>
<p>Some ideas others have had for decentralized institutions in general include:</p>
<ul>
<li><a href="http://trustdavis.io/">TrustDavis</a> (adding skin-in-the-game to e-commerce reputations by making e-commerce ratings <em>be</em> offers to insure others against the receiver of the rating committing fraud)</li>
<li><a href="https://joincircles.net/">Circles</a> (decentralized basic income through locally fungible coin issuance)</li>
<li>Markets for CAPTCHA services</li>
<li>Digitized peer to peer rotating savings and credit <a href="https://www.wetrust.io/">associations</a></li>
<li><a href="https://medium.com/@ilovebagels/token-curated-registries-1-0-61a232f8dac7">Token curated registries</a></li>
<li><a href="https://medium.com/@edmundedgar/snopes-meets-mechanical-turk-announcing-reality-check-a-crowd-sourced-smart-contract-oracle-551d03468177">Crowdsourced smart contract truth oracles</a></li>
<li>Using blockchain-based smart contracts to coordinate unions</li>
</ul>
<p>I would be interested in hearing Posner and Weyl’s opinion on these kinds of “radical markets”, that groups of people can spin up and start using by themselves without requiring potentially contentious society-wide changes to political and property rights. Could decentralized institutions like these be used to solve the key defining challenges of the twenty first century: promoting beneficial scientific progress, developing informational public goods, reducing global wealth inequality, and the big meta-problem behind fake news, government-driven and corporate-driven social media censorship, and regulation of cryptocurrency products: how do we do quality assurance in an open society?</p>
<p>All in all, I highly recommend <em>Radical Markets</em> (and by the way I also recommend Eliezer Yudkowsky’s <em><a href="https://equilibriabook.com/">Inadequate Equilibria</a></em>) to anyone interested in these kinds of issues, and look forward to seeing the discussion that the book generates.</p>
Fri, 20 Apr 2018 18:03:10 -0700
https://vitalik.ca/general/2018/04/20/radical_markets.html
https://vitalik.ca/general/2018/04/20/radical_markets.htmlgeneralGovernance, Part 2: Plutocracy Is Still Bad<p>Coin holder voting, both for governance of technical features, and for more extensive use cases like deciding who runs validator nodes and who receives money from development bounty funds, is unfortunately continuing to be popular, and so it seems worthwhile for me to write another post explaining why I (and <a href="https://medium.com/@Vlad_Zamfir/against-on-chain-governance-a4ceacd040ca">Vlad Zamfir</a> and others) do not consider it wise for Ethereum (or really, any base-layer blockchain) to start adopting these kinds of mechanisms in a tightly coupled form in any significant way.</p>
<p>I wrote about the issues with tightly coupled voting <a href="https://vitalik.ca/general/2017/12/17/voting.html">in a blog post</a> last year, that focused on theoretical issues as well as focusing on some practical issues experienced by voting systems over the previous two years. Now, the latest scandal in DPOS land seems to be substantially worse. Because the delegate rewards in EOS are now so high (5% annual inflation, about $400m per year), the competition on who gets to run nodes has essentially become yet another frontier of US-China geopolitical economic warfare.</p>
<center><img src="https://pic4.zhimg.com/v2-a4b7403626be584f21d47837190e99e0_1200x500.jpg" style="width:400px" /></center>
<p>And that’s not my own interpretation; I quote from <a href="https://zhuanlan.zhihu.com/p/34902188">this article (original in Chinese)</a>:</p>
<blockquote>
<p><strong>EOS supernode voting: multibillion-dollar profits leading to crypto community inter-country warfare</strong></p>
</blockquote>
<blockquote>
<p>Looking at community recognition, Chinese nodes feel much less represented in the community than US and Korea. Since the EOS.IO official Twitter account was founded, there has never been any interaction with the mainland Chinese EOS community. For a listing of the EOS officially promoted events and interactions with communities see the picture below.</p>
</blockquote>
<center><img src="http://vitalik.ca/files/plutocracy_image1.png" style="width:400px" /></center>
<blockquote>
<p>With no support from the developer community, facing competition from Korea, the Chinese EOS supernodes have invented a new strategy: buying votes.</p>
</blockquote>
<p>The article then continues to describe further strategies, like forming “alliances” that all vote (or buy votes) for each other.</p>
<p>Of course, it does not matter at all who the specific actors are that are buying votes or forming cartels; this time it’s some Chinese pools, <a href="https://liskgdt.net/">last time</a> it was “members located in the USA, Russia, India, Germany, Canada, Italy, Portugal and many other countries from around the globe”, next time it could be totally anonymous, or run out of a smartphone snuck into Trendon Shavers’s prison cell. What matters is that blockchains and cryptocurrency, originally founded in a vision of using technology to escape from the failures of human politics, have essentially all but replicated it. Crypto is a reflection of the world at large.</p>
<p>The EOS New York community’s response seems to be that they have issued a strongly worded letter to the world stating that <a href="https://steemit.com/eos/@eosnewyork/block-one-confirms-vote-buying-will-be-against-eos-io-proposed-constitution">buying votes will be against the constitution</a>. Hmm, what other major political entity has <a href="https://en.wikipedia.org/wiki/Emoluments_Clause">made accepting bribes a violation of the constitution</a>? And how has that been going for them lately?</p>
<p><br /></p>
<hr />
<p><br /></p>
<p>The second part of this article will involve me, an armchair economist, hopefully convincing you, the reader, that yes, bribery is, in fact, bad. There are actually people who dispute this claim; the usual argument has something to do with market efficiency, as in “isn’t this good, because it means that the nodes that win will be the nodes that can be the cheapest, taking the least money for themselves and their expenses and giving the rest back to the community?” The answer is, kinda yes, but in a way that’s centralizing and vulnerable to rent-seeking cartels and explicitly contradicts many of the explicit promises made by most DPOS proponents along the way.</p>
<p>Let us create a toy economic model as follows. There are a number of people all of which are running to be delegates. The delegate slot gives a reward of $100 per period, and candidates promise to share some portion of that as a bribe, equally split among all of their voters. The actual N delegates (eg. N = 35) in any period are the N delegates that received the most votes; that is, during every period a threshold of votes emerges where if you get more votes than that threshold you are a delegate, if you get less you are not, and the threshold is set so that N delegates are above the threshold.</p>
<p>We expect that voters vote for the candidate that gives them the highest expected bribe. Suppose that all candidates start off by sharing 1%; that is, equally splitting $1 among all of their voters. Then, if some candidate becomes a delegate with K voters, each voter gets a payment of 1/K. The candidate that it’s most profitable to vote for is a candidate that’s expected to be in the top N, but is expected to earn the fewest votes within that set. Thus, we expect votes to be fairly evenly split among 35 delegates.</p>
<p>Now, some candidates will want to secure their position by sharing more; by sharing 2%, you are likely to get twice as many votes as those that share 1%, as that’s the equilibrium point where voting for you has the same payout as voting for anyone else. The extra guarantee of being elected that this gives is definitely worth losing an additional 1% of your revenue when you do get elected. We can expect delegates to bid up their bribes and eventually share something close to 100% of their revenue. So the outcome seems to be that the delegate payouts are largely simply returned to voters, making the delegate payout mechanism close to meaningless.</p>
<p>But it gets worse. At this point, there’s an incentive for delegates to form alliances (aka political parties, aka cartels) to coordinate their share percentages; this reduces losses to the cartel from chaotic competition that accidentally leads to some delegates not getting enough votes. Once a cartel is in place, it can start bringing its share percentages down, as dislodging it is a hard coordination problem: if a cartel offers 80%, then a new entrant offers 90%, then to a voter, seeking a share of that extra 10% is not worth the risk of either (i) voting for someone who gets insufficient votes and does not pay rewards, or (ii) voting for someone who gets too many votes and so pays out a reward that’s excessively diluted.</p>
<center><img src="http://vitalik.ca/files/plutocracy_image2.png" /></center>
<p><small><i>Sidenote: <a href="https://bitshares.org/technology/delegated-proof-of-stake-consensus/">Bitshares DPOS</a> used approval voting, where you can vote for as many candidates as you want; it should be pretty obvious that with even slight bribery, the equilibrium there is that everyone just votes for everyone.</i></small></p>
<p>Furthermore, even if cartel mechanics <em>don’t</em> come into play, there is a further issue. This equilibrium of coin holders voting for whoever gives them the most bribes, or a cartel that has become an entrenched rent seeker, contradicts explicit promises made by DPOS proponents.</p>
<p>Quoting “<a href="https://hackernoon.com/explain-delegated-proof-of-stake-like-im-5-888b2a74897d">Explain Delegated Proof of Stake Like I’m 5</a>”:</p>
<blockquote>
<p>If a Witness starts acting like an asshole, or stops doing a quality job securing the network, people in the community can remove their votes, essentially firing the bad actor. Voting is always ongoing.</p>
</blockquote>
<p>From “<a href="https://eos.io/documents/EOS_An_Introduction.pdf">EOS: An Introduction</a>”:</p>
<blockquote>
<p>By custom, we suggest that the bulk of the value be returned to the community for the common good - software improvements, dispute resolution, and the like can be entertained. In the spirit of “eating our own dogfood,” the design envisages that the community votes on a set of open entry contracts that act like “foundations” for the benefit of the community. Known as Community Benefit Contracts, the mechanism highlights the importance of DPOS as enabling direct on-chain governance by the community (below).</p>
</blockquote>
<p>The flaw in all of this, of course, is that the average voter has only a very small chance of impacting which delegates get selected, and so they only have a very small incentive to vote based on any of these high-minded and lofty goals; rather, their incentive is to vote for whoever offers the highest and most reliable bribe. Attacking is easy. If a cartel equilibrium does not form, then an attacker can simply offer a share percentage slightly higher than 100% (perhaps using fee sharing or some kind of “starter promotion” as justification), capture the majority of delegate positions, and then start an attack. If they get removed from the delegate position via a hard fork, they can simply restart the attack again with a different identity.</p>
<p><br /></p>
<hr />
<p><br /></p>
<p>The above is not intended purely as a criticism of DPOS consensus or its use in any specific blockchain. Rather, the critique reaches much further. There has been a large number of projects recently that extol the virtues of extensive on-chain governance, where on-chain coin holder voting can be used not just to vote on protocol features, but also to control a bounty fund. Quoting a <a href="https://medium.com/@FEhrsam/blockchain-governance-programming-our-future-c3bfe30f2d74">blog post from last year</a>:</p>
<blockquote>
<p>Anyone can submit a change to the governance structure in the form of a code update. An on-chain vote occurs, and if passed, the update makes its way on to a test network. After a period of time on the test network, a confirmation vote occurs, at which point the change goes live on the main network. They call this concept a “self-amending ledger”.<br />
Such a system is interesting because it shifts power towards users and away from the more centralized group of developers and miners. On the developer side, anyone can submit a change, and most importantly, everyone has an economic incentive to do it. Contributions are rewarded by the community with newly minted tokens through inflation funding. This shifts from the current Bitcoin and Ethereum dynamics where a new developer has little incentive to evolve the protocol, thus power tends to concentrate amongst the existing developers, to one where everyone has equal earning power.</p>
</blockquote>
<p>In practice, of course, what this can easily lead to is funds that offer kickbacks to users who vote for them, leading to the exact scenario that we saw above with DPOS delegates. In the best case, the funds will simply be returned to voters, giving coin holders an interest rate that cancels out the inflation, and in the worst case, some portion of the inflation will get captured as economic rent by a cartel.</p>
<p>Note also that the above is not a criticism of <em>all</em> on-chain voting; it does not rule out systems like futarchy. However, futarchy is untested, but coin voting <em>is</em> tested, and so far it seems to lead to a high risk of economic or political failure of some kind - far too high a risk for a platform that seeks to be an economic base layer for development of decentralized applications and institutions.</p>
<p><br /></p>
<hr />
<p><br /></p>
<p>So what’s the alternative? The answer is what we’ve been saying all along: <em>cryptoeconomics</em>. <a href="https://www.coindesk.com/making-sense-cryptoeconomics/">Cryptoeconomics</a> is fundamentally about the use of economic incentives together with cryptography to design and secure different kinds of systems and applications, including consensus protocols. The goal is simple: to be able to measure the security of a system (that is, the cost of breaking the system or causing it to violate certain guarantees) in dollars. Traditionally, the security of systems often depends on <em>social</em> trust assumptions: the system works if 2 of 3 of Alice, Bob and Charlie are honest, and we trust Alice, Bob and Charlie to be honest because I know Alice and she’s a nice girl, Bob registered with FINCEN and has a money transmitter license, and Charlie has run a successful business for three years and wears a suit.</p>
<p>Social trust assumptions can work well in many contexts, but they are difficult to universalize; what is trusted in one country or one company or one political tribe may not be trusted in others. They are also difficult to quantify; how much money does it take to manipulate social media to favor some particular delegate in a vote? Social trust assumptions seem secure and controllable, in the sense that “people” are in charge, but in reality they can be manipulated by economic incentives in all sorts of ways.</p>
<p>Cryptoeconomics is about trying to reduce social trust assumptions by creating systems where we introduce explicit economic incentives for good behavior and economic penalties for ban behavior, and making mathematical proofs of the form “in order for guarantee X to be violated, at least these people need to misbehave in this way, which means the minimum amount of penalties or foregone revenue that the participants suffer is Y”. <a href="http://arxiv.org/abs/1710.09437">Casper</a> <a href="https://github.com/ethereum/cbc-casper/wiki">is</a> <a href="https://medium.com/@jonchoi/ethereum-casper-101-7a851a4f1eb0">designed</a> to accomplish precisely this objective in the context of proof of stake consensus. Yes, this does mean that you can’t create a “blockchain” by concentrating the consensus validation into 20 uber-powerful “supernodes” and you have to <a href="https://medium.com/@icebearhww/ethereum-sharding-workshop-in-taipei-a44c0db8b8d9">actually think</a> to make a design that intelligently breaks through and navigates existing tradeoffs and achieves massive scalability in a still-decentralized network. But the reward is that you don’t get a network that’s constantly liable to breaking in half or becoming economically captured by unpredictable political forces.</p>
<p><br /></p>
<hr />
<p><br /></p>
<ol>
<li><small><i>It has been brought to my attention that EOS may be reducing its delegate rewards from 5% per year to 1% per year. Needless to say, this doesn't really change the fundamental validity of any of the arguments; the only result of this would be 5x less rent extraction potential at the cost of a 5x reduction to the cost of attacking the system.</i></small></li>
<li><small><i>Some have asked: but how can it be wrong for DPOS delegates to bribe voters, when it is perfectly legitimate for mining and stake pools to give 99% of their revenues back to their participants? The answer should be clear: in PoW and PoS, it's the protocol's role to determine the rewards that miners and validators get, based on the miners and validators' observed performance, and the fact that miners and validators that are pools pass along the rewards (and penalties!) to their participants gives the participants an incentive to participate in good pools. In DPOS, the reward is constant, and it's the voters' role to vote for pools that have good performance, but with the key flaw that there is no mechanism to actually encourage voters to vote in that way instead of just voting for whoever gives them the most money without taking performance into account. Penalties in DPOS do not exist, and are certainly not passed on to voters, so voters have no "skin in the game" (penalties in Casper pools, on the other hand, <b>do</b> get passed on to participants).</i></small></li>
</ol>
Wed, 28 Mar 2018 18:03:10 -0700
https://vitalik.ca/general/2018/03/28/plutocracy.html
https://vitalik.ca/general/2018/03/28/plutocracy.htmlgeneral