Algebraic immunities of vector-valued functions over finite fields

Algebraic immunity is an important cryptographic property of Boolean functions. The notion of algebraic immunity of Boolean functions has been generalized in several ways to vector-valued functions over arbitrary finite fields. In this paper, the results of Ref. [25] are generalized to arbitrary finite fields. We obtain vector-valued functions over arbitrary finite fields such that their algebraic immunities can reach the upper bounds. Furthermore, all the component functions, together with their some nonzero linear combinations, of vector-valued Boolean functions achieved by this construction have optimal algebraic immunities simultaneously.