International Conference on the Theory and Application of Cryptology and Information Security

ASIACRYPT 2002: Advances in Cryptology — ASIACRYPT 2002 pp 267-287

Cryptanalysis of Block Ciphers with Overdefined Systems of Equations

  • Nicolas T. Courtois
  • Josef Pieprzyk
Conference paper

DOI: 10.1007/3-540-36178-2_17

Volume 2501 of the book series Lecture Notes in Computer Science (LNCS)

Abstract

Several recently proposed ciphers, for example Rijndael and Serpent, are built with layers of small S-boxes interconnected by linear key-dependent layers. Their security relies on the fact, that the classical methods of cryptanalysis (e.g. linear or differential attacks) are based on probabilistic characteristics, which makes their security grow exponentially with the number of rounds Nrr.

In this paper we study the security of such ciphers under an additional hypothesis: the S-box can be described by an overdefined system of algebraic equations (true with probability 1). We show that this is true for both Serpent (due to a small size of S-boxes) and Rijndael (due to unexpected algebraic properties). We study general methods known for solving overdefined systems of equations, such as XL from Eurocrypt’00, and show their inefficiency. Then we introduce a new method called XSL that uses the sparsity of the equations and their specific structure.

The XSL attack uses only relations true with probability 1, and thus the security does not have to grow exponentially in the number of rounds. XSL has a parameter P, and from our estimations is seems that P should be a constant or grow very slowly with the number of rounds. The XSL attack would then be polynomial (or subexponential) in Nr>, with a huge constant that is double-exponential in the size of the S-box. The exact complexity of such attacks is not known due to the redundant equations. Though the presented version of the XSL attack always gives always more than the exhaustive search for Rijndael, it seems to (marginally) break 256-bit Serpent. We suggest a new criterion for design of S-boxes in block ciphers: they should not be describable by a system of polynomial equations that is too small or too overdefined.

Key Words

Block ciphersAESRijndaelSquareSerpentCamelliamultivariate quadratic equationsMQ problemoverdefined systems of multivariate equationsXL algorithmGröbner basessparse multivariate polynomialsMultivariate Cryptanalysis
Download to read the full conference paper text

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Nicolas T. Courtois
    • 1
  • Josef Pieprzyk
    • 2
  1. 1.CP8 Crypto Lab, SchlumbergerSemaLouveciennes CedexFrance
  2. 2.Center for Advanced Computing - Algorithms and Cryptography, Department of ComputingMacquarie UniversitySydneyAustralia