Probabilistic Multivariate Cryptography

  • Aline Gouget
  • Jacques Patarin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4341)


In public key schemes based on multivariate cryptography, the public key is a finite set of m (generally quadratic) polynomial equations and the private key is a trapdoor allowing the owner of the private key to invert the public key. In existing schemes, a signature or an answer to an authentication is valid if all the m equations of the public key are satisfied. In this paper, we study the idea of probabilistic multivariate cryptography, i.e., a signature or an authentication value is valid when at least α equations of the m equations of the public key are satisfied, where α is a fixed parameter of the scheme. We show that many new public key signature and authentication schemes can be built using this concept. We apply this concept on some known multivariate schemes and we show how it can improve the security of the schemes.


Signature Scheme Authentication Scheme Authentication Protocol Probabilistic Scheme Cryptology ePrint Archive 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bringer, J., Chabanne, H., Dottax, E.: Perturbing and Protecting a Traceable Block Cipher. Cryptology ePrint Archive, Report 2006/064 (2006)Google Scholar
  2. 2.
    Courtois, N.: The Security of Hidden Field Equations (HFE). In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 266–281. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  3. 3.
    Delsarte, P., Desmedt, Y., Odlyzko, A.M., Piret, P.: Fast Cryptanalysis of the Matsumoto-Imai Public Key Scheme. In: Beth, T., Cot, N., Ingemarsson, I. (eds.) EUROCRYPT 1984. LNCS, vol. 209, pp. 142–149. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    Faugère, J.-C.: A new efficient algorithm for computing Grobner basis (F4). Journal of Pure and Applied Algebra, 61–88 (1999)Google Scholar
  6. 6.
    Faugère, J.-C.: A new efficient algorithm for computing Grobner basis without reduction to zero (F5). In: Proceedings of ISSAC, pp. 75–83. ACM Press, New York (2002)Google Scholar
  7. 7.
    Fell, H., Diffie, W.: Analysis of a public key approach based on polynomial substitution. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 340–349. Springer, Heidelberg (1986)Google Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Imai, H., Matsumoto, T.: Algebraic Methods for Constructing Asymetric Cryptosystems. In: Calmet, J. (ed.) AAECC 1985. LNCS, vol. 229, pp. 108–119. Springer, Heidelberg (1986)Google Scholar
  10. 10.
    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
  11. 11.
    Kipnis, A., Shamir, A.: Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999)Google Scholar
  12. 12.
    Kipnis, A., Shamir, 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
  13. 13.
    Lidl, R., Niederreiter, H.: Finite fields. Encyclopedia of Mathematics and its applications, vol. 20. Cambridge University Press, Cambridge (1997)Google Scholar
  14. 14.
    MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. Elsevier, North-Holl. (1977)MATHGoogle Scholar
  15. 15.
    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)Google Scholar
  16. 16.
    Patarin, J.: Asymmetric Cryptography with a Hidden Monomial. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 45–60. Springer, Heidelberg (1996)Google Scholar
  17. 17.
    Patarin, J.: Cryptanalysis 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 Signature Scheme. In: The Dagstuhl Workshop on Cryptography (1997)Google Scholar
  20. 20.
    Patarin, J., Courtois, N., Goubin, L.: FLASH, a Fast Multivariate Signature Algorithm. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 298–307. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  21. 21.
    Patarin, J., Courtois, N., Goubin, L.: QUARTZ, 128-Bit Long Digital Signatures. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 282–297. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. 22.
    Patarin, J., Goubin, L., Courtois, N.T.: \(C^{*}_{-+}\) 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
  23. 23.
    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
  24. 24.
    Wolf, C., Braeken, A., Preneel, B.: Efficient cryptanalysis of RSE(2)PKC and RSSE(2)PKC. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 145–151. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  25. 25.
    Wolf, C., Preneel, B.: Taxonomy of Public Key Schemes based on the problem of Multivariate Quadratic equations. Cryptology ePrint Archive, Report 2005/077Google Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Aline Gouget
    • 1
  • Jacques Patarin
    • 2
  1. 1.GemaltoIssy-les-MoulineauxFrance
  2. 2.University of VersaillesVersaillesFrance

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