Relative difference sets, graphs and inequivalence of functions between groups

Abstract

For cryptographic purposes, we want to find functions with both low differential uniformity and dissimilarity to all linear functions and to know when such functions are essentially different. For vectorial Boolean functions, extended affine equivalence and the coarser Carlet-Charpin-Zinoviev (CCZ) equivalence are both used to distinguish between nonlinear functions. It remains hard to tell when CCZ equivalent functions are EA-inequivalent. This paper presents a framework for solving this problem in full generality, for functions between arbitrary finite groups. This common framework is based on relative difference sets (RDSs). The CCZ and EA equivalence classes of perfect nonlinear (PN) functions are each derived, by quite different processes, from equivalence classes of splitting semiregular RDSs. By generalizing these processes, we obtain a much strengthened formula for all the graph equivalences which define the EA equivalence class of a given function, amongst those which define its CCZ equivalence class. q

Equivalence classes of functions between finite groups

“…Since f · r ∈ b(f ) by (10), and (f · r) α = (f α ) · ρ(r) by [11,Lemma 15], where ρ is the permutation defined in (13) from α, shift action is wholly confined to bundles. Hence when considering how graph bundles partition into bundles we may ignore any translation action on graphs.…”

“…New material in it is the definition of the three conditions WeakC1(α), C1(α) and C2(α); and the proof of some additional equivalences for [11,Theorem 4] (see Theorem 1).…”

“…For example, when G = N is abelian, the trivial homomorphism 1 : [11,Theorem 21]. In this case, if p = 2 and n is odd, consider the permutation…”

“…In [11], the author generalises these two types of equivalence to functions f : G → N between arbitrary finite groups G and N , and shows that it is sufficient to work with the group C 1 (G, N ) of normalised functions 2 (i.e. with f (1) = 1).…”

“…In previous work [11], the author shows that in the general case of functions f : G → N between arbitrary finite groups G and N , bundle and graph equivalence have a common source in the equivalence relation for splitting semiregular relative difference sets (RDSs). Furthermore, graph equivalent functions f and f are related by a formula…”