Algebraic attacks on cipher systems

Recently, algebraic attacks have emerged as a new type of cryptographic attack on block ciphers. These attacks involve the formulation of a system of nonlinear equations that describe the secret keybits in terms of the input and output bits of the block cipher, which is then solved. Courtois and Pieprzyk have shown that that the AES-Rijndael block cipher can be written as an overdefined system of 8000 multivariate quadratic (MQ) equations with 1600 binary (2002). The estimated complexity for solving these equations requires O(2²³⁰) steps. Recently, Murphy and Robshaw (2002) have shown how to embed Rijndael in a new extended 128-bit cipher, called BES (big encryption system). Within this simplified framework AES encryption can be described by a small and extremely sparse multivariate quadratic (MQ) system over GF(2⁸). Key recovery requires the solution of a sparse MQ systems over GF(2⁸) in approximately O(2¹⁰⁸) steps. This work presents a comparison of these two similar algebraic attacks and a detailed overview of the XL algorithm for solving MQ equations.