Abstract
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 [Formula: see text] 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 [Formula: see text] with optimum algebraic degree, good nonlinearity and good algebraic immunity for all cases [Formula: see text], 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.
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