Forkable Strings are Rare

In a blockchain protocol such as Bitcoin, the users see the world as a sequence of states. A simple yet functional view of this world, for the purpose of analysis, is a Boolean string w = w_1, w_2, \cdots of zeros and ones, where each bit is independently biased towards 1 favoring the “bad guys.”

A bad guy is activated when w_t = 1 for some t. He may try to present the good guys with a conflicting view of the world, such as presenting multiple candidate blockchains of equal length. This view is called a “fork”. A string w that allows the bad guy to fork (with nonnegligible probability) is called a “forkable string”. Naturally, we would like to show that forkable strings are rare: that the manipulative power of the bad guys over the good guys is negligible.

Claim ([1], Bound 2). Suppose w =w_1, \cdots, w_n is a Boolean string, with every bit independently set to 1 with probability (1-\epsilon)/2 for some \epsilon < 1. The probability that w is forkable is at most \exp(-\epsilon^3n/2).

In this post, we present a commentary on the proof that forkable strings are rare. I like the proof because it uses simple facts about random walks, generating functions, and stochastic domination to bound an apparently difficult random process.

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Some Facts about the Gambler’s Ruin Problem

Consider a random walk in the two-dimensional discrete space, where the horizontal direction is indexed by nonnegative time steps and the vertical direction is indexed by integers.

A particle is at its initial position z > 0 at time 0. At every time step, it independently takes a step up or down: up with probability p and down with probability q = 1-p. If p = q the walk is called symmetric. Let p_z and q_z be the probabilities that the particle ever reaches some level a >z or 0, respectively. When it does, the walk stops. The quantity q_z is the ruin probability. We set the initial conditions as q_0 = 1, q_a = 0.

The Gambler’s Ruin Problem: What is the probability that a walk, starting at some z > 0, eventually reaches the origin?

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A Publicly Verifiable Secret Sharing Scheme

We all know Shamir’s (t,n)-Secret Sharing Scheme: A dealer fixes a group \mathbf{Z}_p and selects a random polynomial P(x) of degree at most t-1 over this group. He sets the constant coefficient equal to your secret \sigma, then distributes n arbitrary evaluations of this polynomial as shares to the n parties. Any subset of t parties can pool their shares and reconstruct the polynomial via Lagrange interpolation.

What if we don’t want to trust the dealer nor the parties? Feldman’s Verifiable Secret Sharing Scheme allows the dealer to publish, in addition to the regular Shamir-shares, a generator g of the group \mathbf{Z}_p^* along with t proofs or commitments c_j for each coefficient of the polynomial P(x). Each party can verify that his share is correct by testing whether

\displaystyle g^{P(i)}\, \overset{?}{=} \, \prod_j{c_j^{i^j}} \bmod p.

However, the shares p(i) are still private. What if we want everyone to be able to verify the correctness of all shares without knowing what the shares are? This is achieved by a Publicly Verifiable Secret Sharing Scheme, such as the one developed by Berry Schoenmaker, assuming the discrete log problem is hard.

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Two MDS Array Codes for Disk Erasures: the Blaum-Bruck-Vardy Code and the BASIC Code

In this post, I am going to review two erasure codes: the Blaum-Bruck-Vardy code and the BASIC code (also here). These are erasure codes, which means, their purpose is to encode a number of data disks into a number of coding disks so that when one or more data/coding disks fail, the failed disk can be reconstructed using the existing data and coding disks.

A strength of these codes is that although the algebra is described on extension fields/rings over GF(2), the encoding/decoding process uses only Boolean addition/rotation operation and no finite field operation. These codes are also MDS (Maximum Distance Separable), which means they have the largest possible (minimum) distance for a fixed message-length and codeword-length.

(Recall that if a code has d data components and c parity components in its generator matrix in standard form, its distance is at most c + 1 by the Singleton bound. Hence the code is MDS if and only if it can tolerate c arbitrary disk failures.)

The BASIC code does the following things in relations to the BBV code:

  1. Adds a virtual parity bit after each disk, giving each disk an even parity
  2. Does polynomial arithmetic modulo 1+x^p instead of h(x) = 1+x+\cdots + x^{p-1} as in the case of BBV code
  3. Shows equivalence to the BBV code by making a nice observation via Chinese Remainder Theorem
  4. Proves MDS property for any number of coding disks when p is “large enough” and has a certain structure

Open Question: What is the least disk size for which these codes are MDS with arbitrary distance?

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Generalized Vandermonde Matrices with Missing Row-Powers

Let [n] := \{0, 1, 2, \cdots, n -1 \}. Let x = (x_0, \cdots, x_{n-1}) with distinc x_js. Let R = \{0 = r_1, r_2, \cdots, r_{n-1} = r be a set of distinct nonnegative row-powers. Consider the n \times n Generalized Vandermonde matrix V(x ; R) := \left( x_j^{r_i} \right)_{i,j \in [n]}. When R = [n], this matrix becomes the ordinary Vandermonde matrix, V(x).

An equivalent description of R is the largest row-power r \in R and the set of missing rows from [r]: that is, the items that are in [r] but not in R. Let L_r = \{\ell_0, \ell_1, \cdots \} be this set. Define the punctured Generalized Vandermonde matrix V_{\perp}(x; L_r) := V(x; R). Let s_k(x) be the kth elementary symmetric polynomial.

Now we are ready to present some interesting results without proof, from the paper “Lower bound on Sequence Complexity” by Kolokotronis, Limniotis, and Kalouptsidis.

det V_{\perp}(x ; \{\ell \}) = det V(x) \ s_{n-\ell}(x).

det V_{\perp}(x ; \{\ell_0, \ell_1\}) = det V(x) \ det \left( s_{n-\ell_i+j}(x) \right)_{i,j \in [2]}.

det V_{\perp}(x ; L) = det V(x) \ det \left( s_{n-\ell_i+j}(x) \right)_{i,j \in [s]} \text{ where } L = \{ \ell_0, \ell_1, \cdots, \ell_{s-1}\}.

I will add some applications later on.

Vandermonde Submatrices and Arithmetic Progressions

[This post, which is based on an ongoing discussion with Alex Russell and Ravi Sundaram, contains some unpublished results.]

Currently, we are asking whether all submatrices of the order-p Vandermonde matrix over a finite extension of GF(2) are invertible where p is prime. The answer is “no” in general: there are examples of fields where the Vandermonde matrix has a singular submatrix.

We can ask an easier(?) question, though. What happens if we randomly sample a set of columns and look into submatrices formed by a subset of the sampled columns. With a touch of beautiful insight, Professor Russell has connected Szemeredi’s theorem on arithmetic progressions with this question.

Let AP_k denote an arithmetic progression of length $latek k$. Let [N] := \{1, 2, \cdots, N\} for N \in \mathbb{N}.

The Szemerédi theorem says, any “sufficiently dense” subset S \subset [N] contains infinitely many AP_k for all k \in \mathbb{N}. A finitary version says: Fix your favourite k \in \mathbb{N}, \delta \in [0, 1]. Then,  there exists a natural N := N_{k, \delta} such that if you look any subset S \subset [N] of size at least \delta N, you will find an AP_k. Yet another version says:

Szemerédi’s Theorem. The size of the largest subset S \subset [N] without an AP_k cannot be too large; in particular, it is o(N).

Recall that a function f(x) is o(g) if it grows too slow compared to g(x), so that \lim_{N\rightarrow \infty}{f(x)/g(x) = 0}.

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When does the Discrete Fourier Transform Matrix have Nonsingular Submatrices?

I am studying a coding theory problem. The question is this:

Open Question: Is there a prime p and a positive integer d such that all submatrices of the p\times p Discrete Fourier Transform matrix over the field GF(2^d) are nonsingular?

Currently, I have only counterexamples: Let d be the degree of the smallest extension over GF(2) which contains a nontrivial pth root of unity. Then, I know a lot of primes p for which the matrix V has a singular submatrix.

In this post, I am going to show a failed attempt to answer this question using the results in this paper by Evra, Kowalski, and Lubotzky.

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Takeaways from “Never Split the Difference” by Chris Voss

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HBO charged me $15 via Amazon for my subscription that I forgot to cancel after the Game of Thrones was over. Upon writing to Amazon, the representative said:

As per policy we are unable to issue refund for the previous month charge (September) for your HBO subscription. However, I am able to make an exception for you and have issued you a promotional credit of $15.14 in your account.

Walmart manager said he would not accept my returns (worth $107) without a receipt.  Although the first guy said he would give me a gift card, the system didn’t let him proceed. The manager came in and told me they wouldn’t take these returns. Why? Because they have recently changed the rule: without a receipt, I can return items worth only up to $25.

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An Intuitive Explanation of the Discrete Fourier Transform

Every time I thought I understand Fourier Transform, I was wrong.

Doing the Fourier Transform of a function is just seeing it from “another” point of view. The “usual” view of a function is in the standard basis \{e_1, \cdots, e_n\}. For example, f can be seen as a vector (in the basis given by the elements in the domain) whose coordinates are the evaluations of f on the elements in the domain. It can also be seen as a polynomial (in the monomial basis) whose coefficients are these evaluations. Let us call this vector u.

The same function can also be seen from the “Fourier basis”, which is just another orthogonal basis, formed by the basis vectors \{v_t\}, t. The tth coordinate in the new basis will be given by inner product between u and the tth basis vector v_t. We call these inner products the Fourier coefficients. The Discrete Fourier Transform matrix (the DFT matrix) “projects” a function from the standard basis to the Fourier basis in the usual sense of projection: taking the inner product along a given direction.

In this post, I am going to use elementary group theoretic notions, polynomials, matrices, and vectors. The ideas in this post will be similar to this Wikipedia article on Discrete Fourier Transform.

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Why x^p-1 Factors over GF(2) the Way It Does

Let p be a prime number. We know that F_2=GF(2)=\{0,1\}. Let F_2[x] be the ring  the polynomials with indeterminate x and coefficients in F_2. The polynomial (x^p-1) \in F_2[x] factors over F_2 as  follows for various p:

x^3-1=(1+x) (1+x+x^2).
x^5-1=(1+x) (1+x+x^2+x^3+x^4).
x^7-1=(1+x) (1+x+x^3) (1+x^2+x^3).
x^{11}-1=(1+x) (1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}).
x^{13}-1=(1+x) (1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}).
x^{17}-1=(1+x) (1+x^3+x^4+x^5+x^8) (1+x+x^2+x^4+x^6+x^7+x^8).

None of the factors above can be factored anymore, hence they are irreducible over F_2. Let us call x+1 the trivial factor since the root x= 1 already belongs to F_2. But why do we have two nontrivial irreducible factors of x^7-1, each of degree 3, whereas x^{13}-1 has only one non-trivial irreducible factor of degree 12? It appears that either there is only one nontrivial factor, or the degree of all nontrivial factors is the same. Why?

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