Advertisement

MXL2: Solving Polynomial Equations over GF(2) Using an Improved Mutant Strategy

  • Mohamed Saied Emam Mohamed
  • Wael Said Abd Elmageed Mohamed
  • Jintai Ding
  • Johannes Buchmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5299)

Abstract

MutantXL is an algorithm for solving systems of polynomial equations that was proposed at SCC 2008. This paper proposes two substantial improvements to this algorithm over GF(2) that result in significantly reduced memory usage. We present experimental results comparing MXL2 to the XL algorithm, the MutantXL algorithm and Magma’s implementation of F 4. For this comparison we have chosen small, randomly generated instances of the MQ problem and quadratic systems derived from HFE instances. In both cases, the largest matrices produced by MXL2 are substantially smaller than the ones produced by MutantXL and XL. Moreover, for a significant number of cases we even see a reduction of the size of the largest matrix when we compare MXL2 against Magma’s F 4 implementation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Matsumoto, T., Imai, H.: Public Quadratic Polynomial-Tuples for Efficient Signature-Verification and Message-Encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  2. 2.
    Patarin, J.: Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of Asymmetric Algorithms. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  3. 3.
    Patarin, J., Goubin, L., Courtois, N.: \(C^{*}_{-+}\) and HM: Variations Around Two Schemes of T. Matsumoto and H. Imai. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 35–50. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  4. 4.
    Moh, T.: A Public Key System With Signature And Master Key Functions. Communications in Algebra 27, 2207–2222 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ding, J.: A New Variant of the Matsumoto-Imai Cryptosystem through Perturbation. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 305–318. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Courtois, N.T., Klimov, A., Patarin, J., Shamir, A.: Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Ding, J., Buchmann, J., Mohamed, M.S.E., Moahmed, W.S.A., Weinmann, R.P.: MutantXL. In: Proceedings of the 1st international conference on Symbolic Computation and Cryptography (SCC 2008), Beijing, China, LMIB, pp. 16–22 (2008), http://www.cdc.informatik.tu-darmstadt.de/reports/reports/MutantXL_Algorithm.pdf
  8. 8.
    Ding, J., Cabarcas, D., Schmidt, D., Buchmann, J., Tohaneanu, S.: Mutant Gröbner Basis Algorithm. In: Proceedings of the 1st international conference on Symbolic Computation and Cryptography (SCC 2008), Beijing, China, LMIB, pp. 23–32 (2008)Google Scholar
  9. 9.
    Courtois, N.T.: Experimental Algebraic Cryptanalysis of Block Ciphers (2007), http://www.cryptosystem.net/aes/toyciphers.html
  10. 10.
    Segers, A.: Algebraic Attacks from a Gröbner Basis Perspective. Master’s thesis, Department of Mathematics and Computing Science, TECHNISCHE UNIVERSITEIT EINDHOVEN, Eindhoven (2004)Google Scholar
  11. 11.
  12. 12.
    Albrecht, M., Bard, G.: M4RI – Linear Algebra over GF(2) (2008), http://m4ri.sagemath.org/index.html

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Mohamed Saied Emam Mohamed
    • 1
  • Wael Said Abd Elmageed Mohamed
    • 1
  • Jintai Ding
    • 2
  • Johannes Buchmann
    • 1
  1. 1.TU Darmstadt, FB InformatikDarmstadtGermany
  2. 2.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations