Monomial bent functions and Stickelberger's theorem
type: Item
instance of: scholarly article
title: Monomial bent functions and Stickelberger's theorem
zbMATH Open document ID: 1159.11052
author: Philippe Langevin
author: Gregor Leander
published in: Finite Fields and their Applications
publication date: 2008-08-27T00:00:00Z
review text: The authors use results on Gauss sums, mainly Stickelberger's theorem, to obtain a necessary and sufficient criterion for certain monomial functions from \(L = \mathbb F_{2^n}\) into \(\{-1,1\}\) of the form \(\mu_L(\alpha x^d)\) with \(\mu_L(x) = (-1)^{\text{tr}(x)}\), \(\text{tr}(x)\) denotes the absolute trace function, to be bent: For integers \(d,k,n\) with \(n = 2k\), consider the mapping \(V_d: \mathbb Z/(2^n-1)\mathbb Z \rightarrow \{0,1,\ldots,2n\}\) given by \(V_d(j) = \text{wt}(j) + \text{wt}(-jd)\), where \(\text{wt}(z)\) is the weight of the binary representation of \(z\) modulo \(2^n-1\). Suppose that the integer \(d\) satisfies the conditions \[ \min_{0 < j < 2^n-1}V_d(j) = k, \quad\text{and}\quad V_d(j) = k \Rightarrow jd = 0, \] then \(\mu_L(\alpha x^d)\) is bent if and only if \(\sum_{j \in \mathcal{I}_d}\alpha^j = 1\), with \(\mathcal{I}_d = \{j\;|\;V_d(j) = k\}\). Furthermore the dual function is then described. Using this result the authors give short alternative proofs for the bentness of the Gold function and the Kasami function. In the latter case the authors do not need the restriction \(\gcd(n,3) = 1\) as it is needed in previous proofs [see \textit{J. F. Dillon} and \textit{H. Dobbertin} [Finite Fields Appl. 10, 342--389 (2004; Zbl 1043.05024)]. Finally the authors show that the dual of the Kasami function as not monomial.
reviewed by: Wilfried Meidl
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94A60
Mathematics Subject Classification ID: 06E30
Mathematics Subject Classification ID: 11T24
zbMATH DE Number: 5316925
zbMATH Keywords: Boolean functions
zbMATH Keywords: bent functions
zbMATH Keywords: Kasami function
zbMATH Keywords: Gold function
zbMATH Keywords: nonlinearity
zbMATH Keywords: Gauss sum
zbMATH Keywords: Stickelberger's congruence
MaRDI profile type: Publication
full work available at URL: j.ffa.2007.12.001
OpenAlex ID: W2025045554
cites work: A new class of monomial bent functions
cites work: Finding nonnormal bent functions
cites work: Cubic Monomial Bent Functions: A Subclass of $\mathcal{M}$
cites work: Q4115154
cites work: New cyclic difference sets with Singer parameters
cites work: Q2712116
cites work: A proof of the Welch and Niho conjectures on cross-correlations of binary \(m\)-sequences
cites work: Q3922765
cites work: Q3880931
cites work: Q3797302
cites work: Monomial bent functions
cites work: Q3216750
cites work: The Gross-Koblitz formula revisited
cites work: On Some Exponential Sums
DOI: 10.1016/J.FFA.2007.12.001
Recommended article: Monomial and quadratic bent functions over the finite fields of odd characteristic
Recommended article: A new class of monomial bent functions
Recommended article: On the Dual of Monomial Quadratic p-ary Bent Functions
Recommended article: Q3635547
Recommended article: Near bent functions on a hyperplane
Recommended article: A new class of bent and hyper-bent Boolean functions in polynomial forms
Recommended article: Gold and Kasami-Welch functions, quadratic forms, and bent functions
Recommended article: Cubic Monomial Bent Functions: A Subclass of $\mathcal{M}$
Recommended article: Constructions of \(p\)-ary quadratic bent functions
Recommended article: Monomial bent functions
description: scientific article; zbMATH DE number 5316925