TY - JOUR

T1 - Bounds on the degree of APN polynomials: the case of x −1 + g(x)

A1 - Leander,Gregor

A1 - Rodier,François

AU - Leander,Gregor

AU - Rodier,François

PY - 2011

Y1 - 2011

N2 - In this paper we consider APN functions $${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$$ of the form f(x) = x −1 + g(x) where g is any non $${\mathcal{F}_{2}}$$-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields $${\mathcal{F}_{2^m}}$$. Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ≤ 3 where these functions are equivalent to x 3.

AB - In this paper we consider APN functions $${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$$ of the form f(x) = x −1 + g(x) where g is any non $${\mathcal{F}_{2}}$$-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields $${\mathcal{F}_{2^m}}$$. Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ≤ 3 where these functions are equivalent to x 3.

U2 - 10.1007/s10623-010-9456-y

DO - 10.1007/s10623-010-9456-y

JO - Designs, Codes and Cryptography

JF - Designs, Codes and Cryptography

SN - 09251022

IS - 1-3

VL - 59

SP - 207

EP - 222

ER -