CCZ and EA equivalence between mappings over finite Abelian groups

Abstract

Abstract. CCZ-and EA-equivalence, which are originally defined for vectorial Boolean functions, are extended to mappings between finite abelian groups G and H. We obtain an extension theorem for CCZequivalent but not EA-equivalent mappings. Recent results in [1] are improved and generalized.

P℘N functions, complete mappings and quasigroup difference sets

“…, are abelian groups and f g , are functions defined from G to H , then we say that f g , are extended affine equivalent (EA-equivalent) [3,19] if there exist two automorphisms…”

“…Recall that if $$G,H$$ are abelian groups and $$f,g$$ are functions defined from $$G$$ to $$H$$, then we say that $$f,g$$ are extended affine equivalent (EA‐equivalent) [3, 19] if there exist two automorphisms $${\phi }_{1}\in \text{Aut}(G)$$, $${\phi }_{2}\in \text{Aut}(H)$$ (the automorphism group of $$G$$, respectively, $$H$$), and $$\psi \in \text{Hom}(G,H)$$ (the group of homomorphisms from $$G$$ to $$H$$), $${c}_{1}\in G,{c}_{2}\in H$$ such that $$g(x)={\phi }_{2}(f({\phi }_{1}(x)+{c}_{1}))+\psi (x)+{c}_{2}$$. If $$\psi =0$$, we recover the affine equivalence .…”

Relationships Between CCZ and EA Equivalence Classes and Corresponding Code Invariants

“…Case 2 follows from Corollary 2 because any automorphism of G × N must fix {1} × N (and G × {1} by symmetry). The argument is due to Pott and Zhou [25] for G abelian but holds in general, and in particular, includes the case…”