Constructing vectorial Boolean functions with high algebraic immunity based on group decomposition

In this paper, we construct a class of vectorial Boolean functions over F2n with high algebraic immunity based on the decomposition of the multiplicative group of F2n . By viewing F2n as G1G2 ∪ {0} (where G1 and G2 are subgroups of F2n , (#G1,#G2) = 1 and #G1 × #G2 = 2 − 1), we give a generalized description for constructing vectorial Boolean functions with high algebraic immunity. Moreover, when n is even, we provide two special classes of vectorial Boolean functions with high(sometimes optimal) algebraic immunity, one is hyper-bent, and the other is of balancedness and optimal algebraic degree .