In this paper we describe an approach for solving complex multivariate equation systems related to algebraic cryptanalysis. The work uses the newly introduced Compressed Right Hand Sides (CRHS) representation, where equations are represented using Binary Decision Diagrams (BDD). The paper introduces a new technique for manipulating a BDD, similar to swapping variables in the well-known sifting-method. Using this technique we develop a new solving method for CRHS equation systems. The new algorithm is successfully tested on systems representing reduced variants of Trivium.