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

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 $t\ge 3$ 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 $\omega ,{\omega }^t$ 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 $4^i-2^i+1$.

Cited by in corpus (12)

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• Functions which are PN on infiitely many extensions of Fp, p odd
• Exceptional Scattered Polynomials
• A new large class of functions not APN infinitely often
• Asymptotics of Moore exponent sets
• Plane sections of Fermat surfaces over finite fields
• Differential properties of functions x -> x^{2^t-1} -- extended version
• On Inversion in Z_{2^n-1}