Recent results on constructing Boolean functions with (potentially) optimal algebraic immunity based on decompositions of finite fields (Q1730316)

In this survey article, the authors review their contributions (with several co-authors) generalizing step by step previous constructions of Boolean functions with optimal algebraic immunity (OAI-functions), see the papers [\textit{Q. Jin} et al., J. Syst. Sci. Complex. 28, No. 3, 725--742 (2015; Zbl 1336.94057); \textit{J. Zheng} et al., Int. J. Found. Comput. Sci. 25, No. 5, 537--551 (2014; Zbl 1370.94549); \textit{B. Wu} et al., ``Constructing Boolean functions with (potentially) optimal algebraic immunity based on multiplicative decompositions of finite fields'', Preprint, \url{arXiv:1401.6604}]. After recalling basic notions and pointing to some previous constructions, in Section 3 the authors describe their generalizations of some earlier constructions based on additive decompositions of a finite field (such as \(\mathbb{F}_{2^n}\equiv\mathbb{F}_{2^m}\times\mathbb{F}_{2^m}\) if \(n=2m\)). Similarly as Tu-Deng functions [\textit{Z. Tu} and \textit{Y. Deng}, Des. Codes Cryptography 60, No. 1, 1--14 (2011; Zbl 1226.94013)], which are OAI-functions if a combinatorial conjecture (Tu-Deng conjecture) holds, the OAI property depends on the correctness of variants or generalization of the Tu-Deng conjecture. In Section 4, the authors describe their generalizations of some earlier constructions based on multiplicative decompositions of a finite field. Some constructions are unconditionally OAI-functions, some again under the assumption of the correctness of a combinatorial conjecture.