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Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions
- Source :
- Journal of Algebra, Volume 343, Issue 1, October 2011, Pages 78-92
- Publication Year :
- 2009
-
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 $\omega, \omega^t$ 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 $4^i-2^i+1$.
- Subjects :
- Computer Science - Information Theory
Mathematics - Algebraic Geometry
Subjects
Details
- Database :
- arXiv
- Journal :
- Journal of Algebra, Volume 343, Issue 1, October 2011, Pages 78-92
- Publication Type :
- Report
- Accession number :
- edsarx.0903.2016
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.jalgebra.2011.06.019