On relations between CCZ- and EA-equivalences

Abstract

In the present paper we introduce some sufficient conditions and a procedure for checking whether, for a given function, CCZ-equivalence is more general than EA-equivalence together with taking inverses of permutations. It is known from [8,6] that for quadratic APN functions (both monomial and polynomial cases) CCZ-equivalence is more general. We prove hereby that for non-quadratic APN functions CCZ-equivalence can be more general (by studying the only known APN function which is CCZ-inequivalent to both power functions and quadratics). On the contrary, we prove that for power non-Gold APN functions, CCZ equivalence coincides with EA-equivalence and inverse transformation for n ≤ 8. We conjecture that this is true for any n.

On the EA-classes of known APN functions in small dimensions

“…In this section we will recall the procedure given in [7] and give some remarks and properties regarding CCZ-equivalence that will be useful in the investigation of the EA-classes contained in a CCZ-class.…”

“…Since we are interested in the EA-classes, without loss of generality, we assume that the affine permutation in the definition of CCZ-equivalence is linear. Indeed, using affine permutations instead of linear one we simply obtain a shift by a constant in the input and output of the resulting function (see for instance [7]). A linear map L defined over (F 2 n ) 2 can be described as a formal matrix…”

“…In [7], the authors introduce a procedure that permits to investigate the relation between CCZ-equivalence and EA-equivalence together with the inverse transformation (when applicable). Using this procedure it is possible, at least in small dimensions, to investigate the EA-classes contained in the CCZ-class of a given function.…”

“…It was shown by Budaghyan et al [3] that for quadratic APN functions CCZ-equivalence is more general than EA-equivalence together with taking inverses of permutations. In [7] the authors investigate further the relation between CCZ-equivalence and EA-equivalence with inverse transformation. While, in [9] the authors give a characterization of CCZ-equivalence in terms of twisting functions.…”

“…In this work, we use the procedure introduced in [7] for investigating the EA-classes contained in a CCZ-class of a given function. In order to do that, in Section 3 we give some propositions that can be used to improve the the procedure given in [7] and filter some of the results obtained from this procedure. We also obtain that the number of EA-classes contained in the CCZ-class of a function F is upper bounded by the number of simplex codes contained in a linear code associated to F .…”