Balanced Boolean Functions with (Almost) Optimal Algebraic Immunity and Very High Nonlinearity

In this paper, we present a class of 2k-variable balanced Boolean functions and a class of 2k-variable 1-resilient Boolean functions for an integer k ≥ 2, which both have the maximal algebraic degree and very high nonlinearity. Based on a newly proposed conjecture by Tu and Deng, it is shown that the proposed balanced Boolean functions have optimal algebraic immunity and the 1-resilient Boolean functions have almost optimal algebraic immunity. Among all the known results of balanced Boolean functions and 1-resilient Boolean functions, our new functions possess the highest nonlinearity. Based on the fact that the conjecture has been verified for all k ≤ 29 by computer, at least we have constructed a class of balanced Boolean functions and a class of 1-resilient Boolean functions with the even number of variables ≤ 58, which are cryptographically optimal or almost optimal in terms of balancedness, algebraic degree, nonlinearity, and algebraic immunity.