On differential equivalence of quadratic apn functions

A vectorial Boolean function F : Fn ^ Fn is called almost perfect nonlinear (APN) if the equation F(x) + F(x + a) = b has at most 2 solutions for all vectors a,b Е Fn, where a is nonzero. For a given F, an associated Boolean function y_f(a,b) in 2n variables is defined so that it takes value 1 iff a is nonzero and the equation F(x) + +F(x+a) = b has solutions. We introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. The problem to describe the differential equivalence class of a given APN function is very interesting since the answer can potentially lead to some new constructions of APN functions. We start analyzing this problem with the consideration of affine functions A such that a quadratic APN function F and F + A are differentially equivalent functions. We completely describe these affine functions A for an arbitrary APN Gold function F. Computational results for known quadratic APN functions in small number of variables (2, ..., 8) are presented.

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