Establishing equations: The complexity of algebraic and fast algebraic attacks revisited

Abstract

Algebraic and fast algebraic attacks have posed serious threats to some deployed LFSR-based stream ciphers. Previous works on this topic focused on reducing the time complexity by lowering the degree of the equations, speeding up the substitution step by Fast Fourier Transform and analysis of Boolean functions exhibiting the optimal algebraic immunity. All of these works shared and overlooked a common base, i.e., establishing an adequate equation system first, which actually in some cases dominates the time or memory complexity if the direct methods are used, especially in fast algebraic attacks. In this paper, we present a complete analysis of the establishing equation procedure and show how the Frobenius form of the monomial state rewriting matrix can be applied to considerably reduce the complexity of this step.