arXiv subjects

Christopher Williamson

Publications and source records attributed to Christopher Williamson.

6 recordsLinked to original sources

arXiv2026

Secret Sharing at the Shannon Ceiling

For every $n\geq 9$ that is a multiple of 3, we construct an explicit access structure on $n$ participants. In every perfect secret-sharing scheme realising this access structure, if $S$ denotes the random secret, then the sum of the share entropies is at least $\left(\frac{n^2}{9}+\frac{2n}{3}\right)H(S)$, and some participant has share entropy at least $\left(\frac{n}{6}+\frac12\right)H(S)$. After normalisation by $H(S)$, these are respectively $\Omega(n^2)$ and $\Omega(n)$ lower bounds and also give the same asymptotic lower bounds on the total and largest expected binary lengths of the shares. This improves by a logarithmic factor the longstanding general lower bounds of $\Omega(n^2/\log n)$ for total share size and $\Omega(n/\log n)$ for maximum share size due to Csirmaz. The proof uses only elementary Shannon inequalities, together with some averaging arguments. The Shannon-information method has universal $O(n^2)$ and $O(n)$ ceilings for the total and maximum normalised entropy lower bounds it can certify, so our construction reaches both ceilings up to constant factors.

cs.CC↗

arXiv2026

Finite Three-Colourable (0,2)-Graphs Are Bipartite

A theorem of Payan says that a cubelike graph cannot have chromatic number exactly three. A nearby question, usually discussed as Payan's finite $(0,2)$-graph question, asks whether a finite graph in which every two distinct vertices have either zero or two common neighbours can have chromatic number exactly three. The finite hypothesis is meaningful: infinite three-chromatic $(0,2)$-graphs can be constructed \cite{Payan1992}. We prove that every finite three-colourable $(0,2)$-graph is bipartite. Thus, no finite $(0,2)$-graph has chromatic number exactly three.

math.CO↗

arXiv2026

Symmetric Bounded Indistinguishability: Hypergeometric Smoothing and Hahn Polynomials

A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,\delta)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $\delta$. Previous works introduce this concept and study how far apart $t$-wise marginals can be under an assumption of $(k,\delta)$-wise indistinguishability. We consider symmetric distributions and obtain a new upper bound that unifies and improves previous bounds and applies across a wider range of parameters. In particular, prior works failed to rule out the existence of constants $0 0$ or when $t/n$ tends to 1. Our approach is to exploit the behaviour of the orthogonal Hahn polynomials under hypergeometric sampling and marginalisation operations. As a secondary contribution, we provide nearly matching upper and lower bounds on the maximum possible distance between a pair of $(k,\delta)$-wise indistinguishable distributions and the nearest pair of $(k,0)$-wise indistinguishable distributions.

cs.CC↗

arXiv2025

Extremal poker hand rankings: why the standard 52 card deck and a 3044 card deck are special

We study poker hand rankings in the partially generalised setting of a deck with $r$ ranks, rather than the typical 13 ranks. We provide the hand rankings for all $r$ and observe some interesting phenomena such as the smallest $r$ such that flushes rank below one-pair hands. Perhaps surprisingly, as $r$ grows without bound, the hand ranking is not stable until $r=761$ (a 3044 card deck). We consider showdown frequency, which is the frequency that a given type of hand is declared by a player at showdown, and make note of counterintuitive instances in which a hand with lower absolute frequency than some other hand nonetheless has a higher showdown frequency. This can be interpreted as a form of Gadbois paradox but in the typical setting of poker without wild cards. Conveniently, the standard deck with 13 ranks turns out to be the smallest deck that avoids a discrepancy between absolute frequency and showdown frequency for all hand types other than having a high card.

math.HO↗

arXiv2021

Sharp indistinguishability bounds from non-uniform approximations

We study the problem of distinguishing between two symmetric probability distributions over $n$ bits by observing $k$ bits of a sample, subject to the constraint that all $k-1$-wise marginal distributions of the two distributions are identical to each other. Previous works of Bogdanov et al. and of Huang and Viola have established approximately tight results on the maximal statistical distance when $k$ is at most a small constant fraction of $n$ and Naor and Shamir gave a tight bound for all $k$ in the case of distinguishing with the OR function. In this work we provide sharp upper and lower bounds on the maximal statistical distance that holds for all $k$. Upper bounds on the statistical distance have typically been obtained by providing uniform low-degree polynomial approximations to certain higher-degree polynomials; the sharpness and wider applicability of our result stems from the construction of suitable non-uniform approximations.

cs.CC↗

arXiv2019

Approximate degree, secret sharing, and concentration phenomena

The $ε$-approximate degree $deg_ε(f)$ of a Boolean function $f$ is the least degree of a real-valued polynomial that approximates $f$ pointwise to error $ε$. The approximate degree of $f$ is at least $k$ iff there exists a pair of probability distributions, also known as a dual polynomial, that are perfectly $k$-wise indistinguishable, but are distinguishable by $f$ with advantage $1 - ε$. Our contributions are: We give a simple new construction of a dual polynomial for the AND function, certifying that $deg_ε(f) \geq Ω(\sqrt{n \log 1/ε})$. This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the $1/3$-approximate degree of any read-once DNF is $Ω(\sqrt{n})$. We show that any pair of symmetric distributions on $n$-bit strings that are perfectly $k$-wise indistinguishable are also statistically $K$-wise indistinguishable with error at most $K^{3/2} \cdot \exp(-Ω(k^2/K))$ for all $k < K < n/64$. This implies that any symmetric function $f$ is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-$K$ coalitions with statistical error $K^{3/2} \exp(-Ω(deg_{1/3}(f)^2/K))$ for all values of $K$ up to $n/64$ simultaneously. Previous secret sharing schemes required that $K$ be determined in advance, and only worked for $f=$ AND. Our analyses draw new connections between approximate degree and concentration phenomena. As a corollary, we show that for any $d < n/64$, any degree $d$ polynomial approximating a symmetric function $f$ to error $1/3$ must have $\ell_1$-norm at least $K^{-3/2} \exp({Ω(deg_{1/3}(f)^2/d)})$, which we also show to be tight for any $d > deg_{1/3}(f)$. These upper and lower bounds were also previously only known in the case $f=$ AND.

cs.CC↗