On differential equivalence of APN functions

For a given vectorial Boolean function F from F2 to itself it was defined an associated Boolean function γF (a, b) in 2n variables by C. Carlet, P. Charpin, V. Zinoviev in 1998 that takes value 1 iff a 6= 0 and equation F (x) +F (x+a) = b has solutions. In this paper we introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. To describe differential equivalence class of a given APN function is an open problem of great interest. We obtained that each quadratic APN function G in n variables, n ≤ 6, that is differentially equivalent to a given quadratic APN function F , is represented as G = F + A, where A is an affine function. For the APN Gold function F (x) = x k+1, where gcd(k, n) = 1, we completely described all affine functions A such that F and F + A are differentially equivalent. This result implies that APN Gold functions F with k = n/2− 1 for n = 4t form the first infinitive family of functions up to EA-equivalence having non-trivial differential equivalence class consisting of more that 2 trivial functions Fc,d(x) = F (x+ c) + d, c, d ∈ F2 .