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Multiple right-hand side (MRHS) equations over finite fields are a relatively new tool useful for algebraic cryptanalysis. The main advantage is in an efficient representation of the cryptographic primitives. The main methods to solve systems of MRHS equations are gluing, that relies on merging equations, and various versions of local reduction, that relies on removing partial solutions. In this paper we present a new algorithm to solve MRHS systems. The core of the algorithm is a transformation of the problem of solving an MRHS equation system into a problem of group factorization. We then provide two alternative algorithms to solve the transformed problem. One of these algorithms provides a further transformation to the well-studied closest vector problem. A corollary of our research is that the solution of the group factorization problem arising during the process of solving an MRHS equation system must be as difficult as the cryptanalysis of a corresponding block cipher described by this MRHS system.

A new method to solve MRHS equation systems and its connection to group factorization