Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions

TL;DR

This work addresses the problem of classifying exceptional exponents $t$ for which $f(x)=x^t$ is APN on infinitely many extensions of $\mathbb{F}_2$, equivalently when the binary cyclic code $C_n^t$ has minimum distance 5 for infinitely many $n$. It links coding theory and cryptography through the polynomials $f_t$ and $g_t$, analyzes singularities and applies Bezout’s theorem to derive irreducibility properties, and splits the main argument into cases based on $t=2^i\ell+1$ with odd $\ell$, proving Conjecture 3' in the case $\gcd(\ell,2^i-1)=\ell$ (and Jedlicka’s case when $\gcd(\ell,2^i-1)<\ell$), thereby establishing the Gold and Kasami-Welch forms $t=2^i+1$ or $4^i-2^i+1$ as the exceptional exponents. A counterexample with $t=205$ demonstrates that related conjectures about irreducibility do not hold in general, highlighting the delicate boundary of the classification. The results refine the connection between $APN$ monomials, cyclic codes, and algebraic-geometry methods, with implications for both code design and cryptographic S-box construction.

Abstract

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect Nonlinear) over $\mathbb{F}_{2^n}$ for infinitely many values of $n$. Equivalently, $t$ is exceptional if the binary cyclic code of length $2^n-1$ with two zeros $ω, ω^t$ has minimum distance 5 for infinitely many values of $n$. The conjecture we prove states that every exceptional number has the form $2^i+1$ or $4^i-2^i+1$.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Corollary 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10