Vectorial bent functions in odd characteristic and their components
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Vectorial bent functions in odd characteristic and their components ˘ 1 · Wilfried Meidl2 · Alexander Pott3 Ayc¸a C¸es¸melioglu Received: 19 September 2019 / Accepted: 18 June 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Bent functions in odd characteristic can be either (weakly) regular or non-weakly regular. Furthermore one can distinguish between dual-bent functions, which are bent functions for which the dual is bent as well, and non-dual bent functions. Whereas a weakly regular bent function always has a bent dual, a non-weakly regular bent function can be either dual-bent or non-dual-bent. The classical constructions (like quadratic bent functions, Maiorana-McFarland or partial spread) yield weakly regular bent functions, but meanwhile one knows constructions of infinite classes of non-weakly regular bent functions of both types, dual-bent and non-dual-bent. In this article we focus on vectorial bent functions in odd characteristic. We first show that most p-ary bent monomials and binomials are actually vectorial constructions. In the second part we give a positive answer to the question if non-weakly regular bent functions can be components of a vectorial bent function. We present the first construction of vectorial bent functions of which the components are nonweakly regular but dual-bent, and the first construction of vectorial bent functions with non-dual-bent components. Keywords Vectorial bent functions Mathematics Subject Classification (2010) 94B25 · 11T71 This article belongs to the Topical Collection: Boolean Functions and Their Applications IV Guest Editors: Lilya Budaghyan and Tor Helleseth Ayc¸a C ¸ es¸melio˘glu
[email protected] Wilfried Meidl [email protected] Alexander Pott [email protected] 1
˙Istanbul Bilgi University, Hacıahmet Mahallesi Pir H¨usamettin Sokak No:20, Beyo˘glu, 34440 ˙Istanbul, Turkey
2
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040-Linz, Austria
3
Otto von Guericke University, Faculty of Mathematics, 39106 Magdeburg, Germany
Cryptography and Communications
1 Introduction Let p be a prime, and Vn be an n-dimensional vector space over the prime field Fp . A function f : Vn → Fp is called a bent function if its Walsh transform f(b) =
f (x)−b,x
p
,
p = e2πi/p ,
x∈Vn
has absolute value p n/2 for all b ∈ Vn , where b, x denotes a (nondegenerate) inner product of Vn (if Vn = Fnp , one may take the conventional dot product, the standard inner product for Vn = Fpn is b, x = Trn (bx), where Trn (z) denotes the absolute trace of z ∈ Fpn ). If p = 2, then f(b) is an integer, hence Boolean bent functions only exist for even ∗ dimensions n. Note that then f(b) = 2n/2 (−1)f (b) for a Boolean function f ∗ , called the dual of f . Bent functions from Vn to Fp , p odd, which we will call p-ary bent functions, exist for even and for odd n. For a p-ary bent function, the Walsh coefficient f(b) at b ∈ Vn of f always satisfies (see [1]
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