Classification of self dual quadratic bent functions. (English) Zbl 1264.06021

Hou, Xiang-Dong

Des. Codes Cryptography 63, No. 2, 183-198 (2012).

Every quadratic function

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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

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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

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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 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

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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

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\(O(2n,\mathbb F_2)\), 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.

Reviewer: Wilfried Meidl (İstanbul)

MSC:

94D10	Boolean functions
06E30	Boolean functions
11E57	Classical groups
11T71	Algebraic coding theory; cryptography (number-theoretic aspects)

Keywords:

alternating matrix; bent function; orthogonal group; quadratic function; self-dual bent function; symplectic group

Citations:

Zbl 1204.94118

References:

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