Cryptographic Properties and Structure of Boolean Functions with Full Algebraic Immunity

Studying Boolean functions with high algebraic immunity (i.e., which can provide some kind of resistance against algebraic attack) has attracted much attention recently. In FSE 2005, Dalai, Gupta and Maitra presented the first construction of Boolean functions achieving maximum possible algebraic immunity. However, the important cryptographic properties, such as algebraic degree and nonlinearity, of the Boolean functions constructed using that method could not be answered, except (by experiment) when the number of variables was small (at most 16). In this paper we solve this problem for every number of variables. Further we study the structure of the construction in detail, and we deduce an algorithm for fast evaluation of the functions, which is crucial for a practical use in stream ciphers