Bent and $${{\mathbb {Z}}}_{2^k}$$ Z 2 k -Bent functions from sp
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Bent and Z2k -Bent functions from spread-like partitions Wilfried Meidl1 · Isabel Pirsic1 Received: 29 January 2020 / Revised: 18 September 2020 / Accepted: 21 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Bent functions from a vector space Vn over F2 of even dimension n = 2m into the cyclic group Z2k , or equivalently, relative difference sets in Vn × Z2k with forbidden subgroup Z2k , can be obtained from spreads of Vn for any k ≤ n/2. In this article, existence and construction of bent functions from Vn to Z2k , which do not come from the spread construction is investigated. A construction of bent functions from Vn into Z2k , k ≤ n/6, (and more generally, into any abelian group of order 2k ) is obtained from partitions of F2m × F2m , which can be seen as a generalization of the Desarguesian spread. As for the spreads, the union of a certain fixed number of sets of these partitions is always the support of a Boolean bent function. Keywords Relative difference set · Bent function · Partial spread · Vectorial bent function · Z2k -bent · Partitions Mathematics Subject Classification 06E30 · 05B10 · 94C10
1 Introduction Let (A, + A ), (B, + B ) be finite abelian groups. A function f from A to B is called a bent function if | (1) χ(x, f (x))| = |A| x∈A
for every character χ of A × B which is nontrivial on B. Alternatively, f : A → B is bent if and only if for all nonzero a ∈ A the function Da f (x) = f (x + A a) − B f (x) is balanced, i.e., every value in B is taken on the same number |A|/|B| times. The graph of f ,
Communicated by K.-U. Schmidt.
B
Wilfried Meidl [email protected] Isabel Pirsic [email protected]
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Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, Linz 4040, Austria
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W. Meidl, I. Pirsic
G = {(x, f (x)) : x ∈ A}, is then a relative difference set in A × B relative to B, see [15]. For background on relative difference sets we refer to [16]. In the classical case, A = Vn and B = Vm are elementary abelian 2-groups, i.e., they are vector spaces of dimension n and m respectively over the prime field F2 . In this case the character sum in (1), called Walsh transform of f at (a, b) ∈ V∗m × Vn , is of the form W f (a, b) = (−1)a, f (x)m ⊕b,xn , x∈Vn
where , k denotes an inner product in Vk , and ⊕ denotes addition modulo 2. If Vk = Fk2 we may use the conventional dot product, the standard inner product in F2k , the finite field of order 2k , is u, vk = Tr k (uv), the absolute trace of uv. A function f : Vn → Vm is bent, if m > 1 also called vectorial bent, if and only if |W f (a, b)| = 2n/2 for all nonzero a ∈ Vm and b ∈ Vn . As is well known, n must then be even and m can be at most n/2. Throughout the article, n = 2m shall always be an even integer. For Boolean bent functions, i.e., bent functions f from Vn to F2 , the dual f ∗ is the Boolean ∗ function defined by W f (1, b) = W f (b) = 2n/2 (−1) f (b) , which is always a Boolean bent function as well. There
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