Advances in Cryptology – CRYPTO 2004

Volume 3152 of the series Lecture Notes in Computer Science pp 1-22

On Multiple Linear Approximations

  • Alex BiryukovAffiliated withDept. ESAT/SCD-COSIC, Katholieke Universiteit Leuven
  • , Christophe De CannièreAffiliated withDept. ESAT/SCD-COSIC, Katholieke Universiteit Leuven
  • , Michaël QuisquaterAffiliated withDept. ESAT/SCD-COSIC, Katholieke Universiteit Leuven


In this paper we study the long standing problem of information extraction from multiple linear approximations. We develop a formal statistical framework for block cipher attacks based on this technique and derive explicit and compact gain formulas for generalized versions of Matsui’s Algorithm 1 and Algorithm 2. The theoretical framework allows both approaches to be treated in a unified way, and predicts significantly improved attack complexities compared to current linear attacks using a single approximation. In order to substantiate the theoretical claims, we benchmarked the attacks against reduced-round versions of DES and observed a clear reduction of the data and time complexities, in almost perfect correspondence with the predictions. The complexities are reduced by several orders of magnitude for Algorithm 1, and the significant improvement in the case of Algorithm 2 suggests that this approach may outperform the currently best attacks on the full DES algorithm.


Linear cryptanalysis multiple linear approximations stochastic systems of linear equations maximum likelihood decoding key-ranking DES AES