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On the differential equivalence of APN functions. (English) Zbl 1459.94180
Summary: C. Carlet et al., Des. Codes Cryptography 15, No. 2, 125-156 (1998; Zbl 0938.94011)] defined the associated Boolean function \(\gamma_{F}(a,b)\) in \(2n\) variables for a given vectorial Boolean function \(F\) from \(\mathbb {F}_{2}^{n}\) to itself. It takes value 1 if \(a\neq\mathbf{0}\) 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, \(n\leq 6\), 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.

94D10 Boolean functions
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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