Vectorial Boolean Functions with Good Cryptographic Properties

In this paper we generalize two remarkable results on cryptographic properties of Boolean functions given by Tu and Deng [8] to the vectorial case. Firstly we construct vectorial bent Boolean functions with good algebraic immunity for all cases 1 ⩽ m ⩽ n, and with maximum algebraic immunity for some cases (n,m). Then by modifying F, we get vectorial balanced functions with optimum algebraic degree, good nonlinearity and good algebraic immunity for all cases , and with maximum algebraic immunity for some cases (n,m). Moreover, while Tu-Deng's results are valid under a combinatorial hypothesis, our results (Theorems 4 and 5) are true without this hypothesis.