Skip to main content
SpringerLink
Log in

Probabilistic Multivariate Cryptography

  • Conference paper
Progress in Cryptology - VIETCRYPT 2006 (VIETCRYPT 2006)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 4341))

Included in the following conference series:

Abstract

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.

This work has been partially financially supported by the European Commission through the IST Program under Contract IST-2002-507932 ECRYPT.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bringer, J., Chabanne, H., Dottax, E.: Perturbing and Protecting a Traceable Block Cipher. Cryptology ePrint Archive, Report 2006/064 (2006)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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. 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. 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)

    Chapter  Google Scholar 

  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. 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. 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. 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. Lidl, R., Niederreiter, H.: Finite fields. Encyclopedia of Mathematics and its applications, vol. 20. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  14. MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. Elsevier, North-Holl. (1977)

    MATH  Google Scholar 

  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. 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. 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. 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. Patarin, J.: The Oil and Vinegar Signature Scheme. In: The Dagstuhl Workshop on Cryptography (1997)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  25. Wolf, C., Preneel, B.: Taxonomy of Public Key Schemes based on the problem of Multivariate Quadratic equations. Cryptology ePrint Archive, Report 2005/077

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Gemalto, 34 rue Guynemer, F-92447, Issy-les-Moulineaux, France

    Aline Gouget

  2. University of Versailles, 45 avenue des Etats-Unis, F-78035, Versailles, France

    Jacques Patarin

Authors
  1. Aline Gouget
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacques Patarin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département d’Informatique, École Normale Supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

    Phong Q. Nguyen

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gouget, A., Patarin, J. (2006). Probabilistic Multivariate Cryptography. In: Nguyen, P.Q. (eds) Progress in Cryptology - VIETCRYPT 2006. VIETCRYPT 2006. Lecture Notes in Computer Science, vol 4341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11958239_1

Download citation

  • DOI: https://doi.org/10.1007/11958239_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68799-3

  • Online ISBN: 978-3-540-68800-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation