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Title:
Balanced Boolean Functions with (Almost) Optimal Algebraic Immunity and Very High Nonlinearity
Authors: Xiaohu Tang, Deng Tang, Xiangyong Zeng, Lei Hu
Abstract: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\ge 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\le 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 \le 58, which are cryptographically optimal or almost optimal in terms of balancedness, algebraic degree, nonlinearity, and algebraic immunity.
ePrint: https://eprint.iacr.org/2010/443
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