Constructing Boolean functions with (potentially) optimal algebraic immunity based on multiplicative decompositions of finite fields

In this paper, we investigate on constructing cryptographically significant Boolean functions with n variables based on decompositions of the multiplicative group of the finite field F_2n of the form F_2n* = U × V, where U and V are cyclic subgroups of F_2n* satisfying (|U|, |V|) = 1. For positive integers s, m and n = 2^sm, we obtain classes of unbalanced functions with optimal algebraic immunity in the cases |U| = 2^m + 1, |V| = (2ⁿ-1)/(2^m+1) and |U| = 2^m-1, |V| = (2ⁿ-1)/(2^m-1), respectively, where in the latter case the optimal algebraic immunity is based on correctness of the Tu-Deng conjecture. Functions belonging to both classes can be modified to be balanced ones with (potentially) optimal algebraic immunity and optimal algebraic degree, and computer experiments show that they also have high nonlinearity and good immunity against fast algebraic attacks. As by-products, variants of the Tu-Deng conjecture and combinatorial results on binary strings in analogy to it are also obtained.