Generalizations of bent functions. A survey

Abstract

Bent functions (Boolean functions with extreme nonlinearity properties) are actively studied for their numerous applications in cryptography, coding theory, and other fields. New statements of problems lead to a large number of generalizations of the bent functions many of which remain little known to the experts in Boolean functions. In this article, we offer a systematic survey of them.

New constructions of low-correlation sequences with high-linear complexity

“…Their definitions are given in the follows. We can see more results about the constructions of bent functions and near bent functions in the papers [14][15].…”

Nonexistence of generalized bent functions from $$\mathbb {Z}_{2}^{n}$$ Z 2 n to $$\mathbb {Z}_{m}$$ Z m

“…One of the simplest examples is Rothaus's bent function f (x) = x 1 x 2 + x 3 x 4 + · · · + x 2t−1 x 2t , where n = 2t. For m = 4 and m = 8, several constructions and existence results on GBFs have been found in [5,[8][9][10][11][12][13]. Generalized bent functions have been widely investigated since they are applied in many fields such as communication theory, cryptography and closely related to coding theory and combinatorial design theory.…”

Constructions ofq-ary functions with good global avalanche characteristics

“…A function with minimum absolute WHT values is called q-ary bent, i.e. f ∈ B n,q is q-ary bent if |W f (u)| = q n/2 for every u ∈ Z n q [9,11,15].…”