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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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##### References:
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