Classification of self dual quadratic bent functions
type: Item
instance of: scholarly article
title: Classification of self dual quadratic bent functions
zbMATH Open document ID: 1264.06021
published in: Designs, Codes and Cryptography
publication date: 2012-03-23T00:00:00Z
review text: Every quadratic function from \(\mathbb F_2^t\) to \(\mathbb F_2\) is of the form \(f(x) = xAx^T+c\) for a unique constant \(c\) and a \(t\times t\) matrix over \(\mathbb F_2\) which is unique modulo \(\Lambda_t(\mathbb F_2)\), the group of the symmetric matrices with all diagonal entries \(0\). The associate alternating matrix of \(f\) is \(A+A^T\). The author shows that the quadratic function \(f(x) = xAx^T+c\) on \(\mathbb F_2^{2n}\) is a self-dual or anti-self-dual bent function if and only if \((A+A^T)^2 = I\) and \((A+A^T)A(A+A^T)+A^T \in \Lambda_{2n}(\mathbb F_2)\). This extends a result of \textit{C. Carlet} et al. [Int. J. Inf. Coding Theory 1, No. 4, 384--399 (2010; Zbl 1204.94118)]. Using this result and the fact that (anti) self-duality is invariant under orthogonal coordinate transformations, the author completely classifies all self-dual and anti-self-dual quadratic Boolean bent functions in \(2n\) variables under the action of the orthogonal group \(O(2n,\mathbb F_2)\), by classifying all \(2n\times 2n\) involutionary matrices over \(\mathbb F_2\) under the action of \(O(2n,\mathbb F_2)\). The sizes of the \(O(2n,\mathbb F_2)\)-orbits of self-dual and anti-self-dual quadratic bent functions are determined explicitly.
reviewed by: Wilfried Meidl
Mathematics Subject Classification ID: 06E30
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 6018303
zbMATH Keywords: alternating matrix
zbMATH Keywords: bent function
zbMATH Keywords: orthogonal group
zbMATH Keywords: quadratic function
zbMATH Keywords: self-dual bent function
zbMATH Keywords: symplectic group
author: Xiang-Dong Hou
MaRDI profile type: Publication
full work available at URL: s10623-011-9544-7
OpenAlex ID: W2094885097
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DOI: 10.1007/S10623-011-9544-7
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description: scientific article; zbMATH DE number 6018303