Block Ciphers: Algebraic Cryptanalysis and Gröbner Bases

Abstract

Block ciphers are one of the most important classes of cryptographic algorithms in current use. Commonly used to provide confidentiality for transmission and storage of information, they encrypt and decrypt blocks of data according to a secret key. Several recently proposed block ciphers (in particular the AES (Daemen and Rijmen in The Design of Rijndael, Springer, Berlin, 2002)) exhibit a highly algebraic structure: their round transformations are based on simple algebraic operations over a finite field of characteristic 2. This has caused an increasing amount of cryptanalytic attention to be directed to the algebraic properties of these ciphers. Of particular interest is the proposal of the so-called algebraic attacks against block ciphers. In these attacks, a cryptanalyst describes the encryption operation as a large set of multivariate polynomial equations, which—once solved—can be used to recover the secret key. Thus the difficulty of solving these systems of equations is directly related to the cipher’s security. As a result computational algebra is becoming an important tool for the cryptanalysis of block ciphers. In this paper we give an overview of block ciphers design and recall some of the work that has been developed in the area of algebraic cryptanalysis. We also consider a few computational and algebraic techniques that could be used in the analysis of block ciphers and discuss possible directions for future work.