The Improvement on Algebraic System of Multivariate Quadratic Equations for Rijndael

According to the algebraic expression of the S-box in Rijndael algorithm, an algebraic system of multivariate quadratic equations over GF(28) are proposed to describe Rijndael. The variables of S boxes are supposed rationally and the relations between these variables are used to establish equations in this paper. The derived system of multivariate quadratic equations is sparse and overdefined. The key recovery of Rijndael can be regarded as a problem of solving this system. By comparing with other parallel systems, this system has fewer terms and variables. So it has a lower complexity while applying the XSL (eXtended Sparse Linearization) technique.