On the differential equivalence of APN functions

Abstract

Carlet et al. (Des. Codes Cryptogr. 15, 125–156, 1998) defined the associated Boolean function γF(a,b) in 2n variables for a given vectorial Boolean function F from $\mathbb {F}_{2}^{n}$ to itself. It takes value 1 if a≠0 and equation F(x) + F(x + a) = b has solutions. This article defines the differentially equivalent functions as vectorial functions having equal associated Boolean functions. It is an open problem of great interest to describe the differential equivalence class for a given Almost Perfect Nonlinear (APN) function. We determined that each quadratic APN function G in n variables, n ≤ 6, that is differentially equivalent to a given quadratic APN function F, can be represented as G = F + A, where A is affine. For the APN Gold function F, we completely described all affine functions A such that F and F + A are differentially equivalent. This result implies that the class of APN Gold functions up to EA-equivalence contains the first infinite family of functions, whose differential equivalence class is non-trivial.