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Several new classes of self-dual bent functions derived from involutions. (English) Zbl 1460.11147
Bent functions are Boolean functions \(\mathbb{F}_2^n\to\mathbb{F}_2\) which are maximum nonlinear, which means \(1=f(x+a)+f(x)\) has \(2^{n-1}\) solutions \(x\) (for all \(a\ne 0\)).
It seems that there are plenty of bent functions (see [S. Mesnager, Bent functions. Fundamentals and results. Cham: Springer (2016; Zbl 1364.94008)]), but only few of them seem to be based on an algebraic construction. In [S. Mesnager, IEEE Trans. Inf. Theory 60, No. 7, 4397–4407 (2014; Zbl 1360.94480)], a new construction method has been introduced using permutations \(\varphi_1, \varphi_2\) and \(\varphi_3\) of \(\mathbb{F}_{2^n}\) such that \(\varphi_1+ \varphi_2 + \varphi_3=\psi\) is a permutation, and, additionally, \(\varphi_1^{-1}+ \varphi_2^{-1} + \varphi_3^{-1}=\psi^{-1}\).
In this paper, the authors find three classes of involutions \(\varphi_1, \varphi_2\) and \(\varphi_3\) which satisfy the above properties, which means \(\varphi_1^{-1}+ \varphi_2^{-1} + \varphi_3^{-1}\) is an involutory permutation. The bent functions constructed from these involutions are self-dual. It is an open problem whether the bent functions constructed here are equivalent to known ones.

MSC:
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94D10 Boolean functions
11T06 Polynomials over finite fields
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