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 F:F(2)(n) -> F(2)(m) 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 F':F(2)(n) -> F(2)(m) with optimum algebraic degree, good nonlinearity and good algebraic immunity for all cases 1 <= m <= n/2, 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.