Phase transition in a system of random sparse boolean equations

Abstract

ABSTRACT. Many problems, including algebraic cryptanalysis, can be transformed
to a problem of solving a (large) system of sparse Boolean equations. In
this article we study 2 algorithms that can be used to remove some redundancy
from such a system: Agreeing, and Syllogism method. Combined with appropriate
guessing strategies, these methods can be used to solve the whole system of
equations. We show that a phase transition occurs in the initial reduction of the
randomly generated system of equations. When the number of (partial) solutions
in each equation of the system is binomially distributed with probability of partial
solution p, the number of partial solutions remaining after the initial reduction
is very low for p's below some threshold pt, on the other hand for p > pt the
reduction only occurs with a quickly diminishing probability.