Partitioning Multivariate Polynomial Equations via Vertex Separators for Algebraic Cryptanalysis and Mathematical Applications

We present a novel approach for solving systems of polynomialequations via graph partitioning. The concept of a variable-sharing graph of asystem of polynomial equations is defined. If such graph is disconnected, thenthe system of equations is actually two separate systems that can be solvedindividually. This can provide a significant speed-up in computing the solutionto the system, but is unlikely to occur either randomly or in applications.However, by deleting a small number of vertices on the graph, the variable-sharing graph could be disconnected in a balanced fashion, and in turn thesystem of polynomial equations are separated into smaller ones of similarsizes. In graph theory terms, this process is equivalent to finding balancedvertex partitions with minimum-weight vertex separators.The techniques of finding these vertex partitions are discussed, and ex-periments are performed to evaluate its practicality for general graphs andsystems of polynomial equations. Applications of this approach to the QUADfamily of stream ciphers, algebraic cryptanalysis of the stream cipher Triv-ium and its variants, as well as some mathematical problems in game theoryand computational algebraic geometry are presented. In each of these cases,the systems of polynomial equations involved are well-suited to our graphpartitioning method, and constructive results are discussed. Mathematics Subject Classification (2000). 05C90, 11T71, 68R10, 94A60, 14G50.