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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.
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. that 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.