Gold and Kasami-Welch functions, quadratic forms, and bent functions
type: Item
instance of: scholarly article
title: Gold and Kasami-Welch functions, quadratic forms, and bent functions
zbMATH Open document ID: 1221.11237
author: Jyrki Lahtonen
author: Gary McGuire
author: Harold N. Ward
published in: Advances in Mathematics of Communications
publication date: 2011-03-10T00:00:00Z
review text: Let \(n\) be odd, \(K={\mathbb F}_{2^n}\), \(Q(x)=\text{Tr}(x^{2^k+1}+x^{2^j+1})\) where \(0\leq j<k\). The authors prove that if \(\gcd(k+j,n)=\gcd(k-j,n)=1\), then the number of zeros of \(Q\) in \(K\) is \[ 2^{n-1}+\left(\frac2n\right)2^{\frac{n-1}{2}}, \] where \(\left(\frac2n\right)\) is the Jacobi symbol. This is used to evaluate the sign of some Walsh transforms in terms of a Jacobi symbol. These results are applied to the Walsh transforms of the Gold and Kasami-Welch functions. Gold functions yield bent functions when restricted to certain hyperplanes.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 94B15
zbMATH DE Number: 5864033
zbMATH Keywords: Walsh-Hadamard transform
zbMATH Keywords: quadratic form
zbMATH Keywords: bent function
MaRDI profile type: Publication
full work available at URL: amc.2007.1.243
OpenAlex ID: W2034168611
DOI: 10.3934/AMC.2007.1.243
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description: scientific article; zbMATH DE number 5864033