Improved Agreeing-Gluing Algorithm

Asymptotical complexity of solving a system of sparse algebraic equations over finite fields is studied here. An equation is called sparse if it depends on a bounded number of variables. Finding efficiently solutions to the system of such equations is an underlying hard problem in the cryptanalysis of modern ciphers. New deterministic Improved Agreeing-Gluing Algorithm is introduced. The expected running time of the algorithm on uniformly random instances of the problem is rigorously estimated. The estimate is at present the best theoretical bound on the complexity of solving average instances of the problem. In particular, this is a significant improvement over those in our earlier papers (Semaev, Des Codes Cryptogr 49:47–60, 2008; Semaev, SIAM J Comput 39:388–409 2009). In sparse Boolean equations a gap between the present worst case and the average time complexity of the problem has significantly increased. We formulate Average Time Complexity Conjecture. If proved that will have far-reaching consequences in the field of cryptanalysis and in computing in general.