Cryptanalysis of the Square Cryptosystems

  • Olivier Billet
  • Gilles Macario-Rat
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5912)

Abstract

Following the cryptanalyses of the encryption scheme HFE and of the signature scheme SFLASH, no serious alternative multivariate cryptosystems remained, except maybe the signature schemes UOV and HFE− −. Recently, two proposals have been made to build highly efficient multivariate cryptosystems around a quadratic internal transformation: the first one is a signature scheme called square-vinegar and the second one is an encryption scheme called square introduced at CT-RSA 2009.

In this paper, we present a total break of both the square-vinegar signature scheme and the square encryption scheme. For the practical parameters proposed by the authors of these cryptosystems, the complexity of our attacks is about 235 operations. All the steps of the attack have been implemented in the Magma computer algebra system and allowed to experimentally assess the results presented in this paper.

References

  1. 1.
    Akkar, M.-L., Courtois, N., Goubin, L., Duteuil, R.: A Fast and Secure Implementation of Sflash. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 267–278. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Baena, J., Clough, C., Ding, J.: Square-vinegar signature scheme. In: Buchmann, J., Ding, J. (eds.) PQCrypto 2008. LNCS, vol. 5299, pp. 17–30. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Clough, C., Baena, J., Ding, J., Yang, B.-Y., Chen, M.-S.: Square, a New Multivariate Encryption Scheme. In: Fischlin, M. (ed.) CT-RSA 2009. LNCS, vol. 5473, pp. 252–264. Springer, Heidelberg (2009)Google Scholar
  5. 5.
    Ding, J., Wolf, C., Yang, B.-Y.: ℓ-Invertible Cycles for Multivariate Quadratic Public Key Cryptography. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 266–281. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Dubois, V., Fouque, P.-A., Shamir, A., Stern, J.: Practical Cryptanalysis of SFLASH. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 1–12. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Faugère, J.-C., Joux, A.: Algebraic Cryptanalysis of Hidden Field Equation (HFE) Cryptosystems using Gröbner Bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44–60. Springer, Heidelberg (2003)Google Scholar
  8. 8.
    Faugère, J.-C., Perret, L.: Polynomial Equivalence Problems: Algorithmic and Theoretical Aspects. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 30–47. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Fouque, P.-A., Granboulan, L., Stern, J.: Differential cryptanalysis for multivariate schemes. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 341–353. Springer, Heidelberg (2005)Google Scholar
  10. 10.
    Fouque, P.-A., Macario-Rat, G., Stern, J.: Key Recovery on Hidden Monomial Multivariate Schemes. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 19–30. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Geiselmann, W., Steinwandt, R., Beth, T.: Attacking the Affine Parts of SFLASH. In: Honary, B. (ed.) Cryptography and Coding 2001. LNCS, vol. 2260, pp. 355–359. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. 12.
    Granboulan, L., Joux, A., Stern, J.: Inverting HFE is Quasipolynomial. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 345–356. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Imai, H., Matsumoto, T.: Algebraic Methods for Constructing Asymmetric Cryptosystems. In: Calmet, J. (ed.) AAECC 1985. LNCS, vol. 229, pp. 108–119. Springer, Heidelberg (1986)Google Scholar
  14. 14.
    Kipnis, A., Patarin, J., Goubin, L.: Unbalanced Oil and Vinegar Signature Schemes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 206–222. Springer, Heidelberg (1999)Google Scholar
  15. 15.
    Lidl, R., Niederreiter, H.: Finite fields. Encyclopedia of mathematics and its applications, vol. 20. Cambridge university press, Cambridge (2003)Google Scholar
  16. 16.
    Matsumoto, T., Imai, H.: Public Quadratic Polynominal Tuples for Efficient Signature Verification and Message Encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988)Google Scholar
  17. 17.
    Patarin, J.: Cryptoanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt 1988. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)Google Scholar
  18. 18.
    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)Google Scholar
  19. 19.
    Patarin, J.: The Oil and Vinegar Algorithm for Signatures. Presented at the Dagsthul Workshop on Cryptography (September 1997)Google Scholar
  20. 20.
    Patarin, J., Goubin, L., Courtois, N.T.: Improved Algorithms for Isomorphisms of Polynomials. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 184–200. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  21. 21.
    Shamir, A.: Efficient Signature Schemes Based on Birational Permutations. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 1–12. Springer, Heidelberg (1994)Google Scholar
  22. 22.
    Shamir, A., Kipnis, A.: Cryptanalysis of the Oil & Vinegar Signature Scheme. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 257–266. Springer, Heidelberg (1998)Google Scholar
  23. 23.
    Shor, P.W.: Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM J. Sci. Stat. Comp. 26, 1484 (1997)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Olivier Billet
    • 1
  • Gilles Macario-Rat
    • 1
  1. 1.Orange LabsIssy-les-MoulineauxFrance

Personalised recommendations