<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 <italic>p</italic>, the number of partial solutions remaining after the initial reduction is very low for <italic>p</italic>’s below some threshold <italic>pt</italic>, on the other hand for <italic>p > pt</italic> the reduction only occurs with a quickly diminishing probability.</p></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.

Phase transition in a system of random sparse Boolean equations

Department of Informatics, University of Bergen, PB 7800, N–5020 Bergen, NORWAY

Pavol Zajac, Department of Applied Information and Information Technology, Slovak University of Technology, Ilkoviˇcova 3, SK–812-19 Bratislava, SLOVAKIA

Tatra Mountains Mathematical Publications

{"article-title":"Phase transition in a system of random sparse Boolean equations"}

Many problems, including algebraic cryptanalysis, can be transformed to a problem of solving a (large) system of sparse Boolean equations. In this article we...

Phase transition in a system of random sparse Boolean equations

Publié en ligne: 12 nov. 2012

Pages: 93 - 105

DOI: https://doi.org/10.2478/v10127-010-0008-7

Mots clés
algebraic cryptanalysis, agreeing, Boolean equations, SAT problem.

This content is open access.

Phase transition in a system of random sparse Boolean equations

Publié en ligne: 12 nov. 2012

Pages: 93 - 105

DOI: https://doi.org/10.2478/v10127-010-0008-7

Mots clésalgebraic cryptanalysis, agreeing, Boolean equations, SAT problem.

This content is open access.

Mots clés
algebraic cryptanalysis, agreeing, Boolean equations, SAT problem.