On the Conjecture on APN Functions
Abstract
An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field $\mathbb{F}$ is called exceptional APN, if it is also APN on infinitely many extensions of $\mathbb{F}$. In this article we consider the most studied case of $\mathbb{F}=\mathbb{F}_{2^n}$. A conjecture of Janwa-Wilson and McGuire-Janwa-Wilson (1993/1996), settled in 2011, was that the only exceptional monomial APN functions are the monomials $x^n$, where $n=2^i+1$ or $n={2^{2i}-2^i+1}$ (the Gold or the Kasami exponents respectively). A subsequent conjecture states that any exceptional APN function is one of the monomials just described. One of our result is that all functions of the form $f(x)=x^{2^k+1}+h(x)$ (for any odd degree $h(x)$, with a mild condition in few cases), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2012
- DOI:
- 10.48550/arXiv.1207.5528
- arXiv:
- arXiv:1207.5528
- Bibcode:
- 2012arXiv1207.5528D
- Keywords:
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- Computer Science - Information Theory;
- Mathematics - Algebraic Geometry;
- Mathematics - Combinatorics
- E-Print:
- 15 pages