On a remarkable property of APN Gold functions

In [13] for a given vectorial Boolean function F from F2 to itself it was defined an associated Boolean function γF (a, b) in 2n variables 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. It is an interesting open problem to describe differential equivalence class of a given APN function. We consider the APN Gold function F (x) = x k+1, where gcd(k, n) = 1, and prove that there exist exactly 2 distinct affine functions A such that F and F +A are differentially equivalent if n = 4t for some t and k = n/2± 1; otherwise the number of such affine functions is equal to 2. This theoretical result and computer calculations obtained show that APN Gold functions for k = n/2 ± 1 and n = 4t are the only functions (except one function in 6 variables) among all known quadratic APN functions in 2, . . . , 8 variables that have more than 2 trivial affine functions Ac,d(x) = F (x) + F (x + c) + d, where c, d ∈ F2 , preserving the associated Boolean function when adding to F .