Bent functions from a finite abelian group into a finite abelian group

Abstract

We introduce the notions of an absolutely non-homomorphic function, a minimal function (farthest from homomorphisms) and a bent function, and prove that the class of bent functions coincides with the class of absolutely non-homomorphic functions, a function is uniquely determined by the distances to homomorphisms with shifts, and that in the primary case the bent functions are absolutely minimal.

Generalizations of bent functions. A survey

“…The functions of the form A → S 1 (C) were already considered by Logachev, Sal'nikov, and Yashchenko (see Definition 4). We have [13,24] Theorem 12. A function f : A → B is a bent function if and only if η z •f for every z = 0 is a bent function in sense of Logachev, Sal'nikov and Yashchenko.…”

“…In 2002, V. I. Solodovnikov proposed [13] the most general approach to algebraic generalizations of bent functions. While presenting his results, we use both the original notation and that of [24] which sometimes seems more convenient.…”

“…Take an arbitrary function f : A → B. We present the following definition of [13] in a slightly different form by introducing normalization factors, but this preserves their meaning.…”

“…Definition 5 (Solodovnikov, 2002). A function f : A → B is called a bent function if | f z (y)| 2 = n for every z ∈ B, z = 0, and arbitrary y ∈ A.…”

Nonlinear functions and difference sets on group actions

“…[4,6,7,[9][10][11]13,14,16,17]). They can be used to construct DES-like cryptosystems that are resistant to differential attacks.…”