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

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