On the Resistance of Boolean Functions Against Algebraic Attacks Using Univariate Polynomial Representation

Abstract

In the past few years, algebraic attacks against stream ciphers with linear feedback function have been significantly improved. As a response to the new attacks, the notion of algebraic immunity of a Boolean function f was introduced, defined as the minimum degree of the annihilators of f and f + 1. An annihilator of f is a nonzero Boolean function g , such that f · g = 0. While several constructions of Boolean functions with optimal algebraic immunity have been proposed, there is no significant progress concerning the resistance against the so-called fast algebraic attacks. In this paper, we provide a framework to assess the resistance of Boolean functions against the new algebraic attacks, including fast algebraic attacks. The analysis is based on the univariate polynomial representation of Boolean functions and necessary and sufficient conditions are presented for a Boolean function to have optimal behavior against all the new algebraic attacks. Finally, we introduce a new infinite family of balanced Boolean functions described by their univariate polynomial representation. By applying the new framework, we prove that all the members of the family have optimal algebraic immunity and we efficiently evaluate their behavior against fast algebraic attacks.