Highly Nonlinear Boolean Functions With Optimal Algebraic Immunity and Good Behavior Against Fast Algebraic Attacks

摘要

Inspired by the previous work of Tu and Deng, we propose two infinite classes of Boolean functions of $2k$ variables where $kgeq 2$. The first class contains unbalanced functions having high algebraic degree and nonlinearity. The functions in the second one are balanced and have maximal algebraic degree and high nonlinearity (as shown by a lower bound that we prove; as a byproduct we also prove a better lower bound on the nonlinearity of the Carlet–Feng function). Thanks to a combinatorial fact, first conjectured by the authors and later proved by Cohen and Flori, we are able to show that they both possess optimal algebraic immunity. It is also checked that, at least for numbers of variables $nleq 16$, functions in both classes have a good behavior against fast algebraic attacks. Compared with the known Boolean functions resisting algebraic attacks and fast algebraic attacks, both of them possess the highest lower bounds on nonlinearity. These bounds are however not enough for ensuring a sufficient nonlinearity for allowing resistance to fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for bounded numbers of variables $(nleq 38)$. Moreover, these values are very good. The infinite class of functions we propose in Construction 2 presents, among all currently known constructions, the best provable tradeoff between all the important cryptographic criteria.