Skip to main content
SpringerLink
Log in

Trapdoor one-way permutations and multivariate polynomials

  • Session 12: Public Key Systems II
  • Conference paper
  • First Online:
Information and Communications Security (ICICS 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1334))

Included in the following conference series:

Abstract

This article is divided into three parts. The first part describes the known candidates of trapdoor one-way permutations. The second part presents a new algorithm, called D *. As we will see, this algorithm is not secure. However, in the third part, D * will be a useful tool to present our new candidate trapdoor one-way permutation, called D **. This candidate is based on properties of multivariate polynomials on finite fields, and has similar characteristics to T. Matsumoto and H. Imai's schemes.

What makes trapdoor one-way permutations particularly interesting is the fact that they immediately provide ciphering, signature, and authentication asymmetric schemes.

Our candidate performs excellently in secret key, and secret key computations can be implemented in low-cost smart-cards, i.e. without co-processors. An extended version of this paper can be obtained from the authors.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Bleichenbacher, W. Bosma, A.K. Lenstra, Some Remarks on Lucas-Based Cryptosystems, Advances in Cryptology, Proceedings of CRYPTO'95, Springer-Verlag, pp. 386–396.

    Google Scholar 

  2. I. Blake, X. Gao, R. Mullin, S. Vanstone, T. Yaghoobian, Applications of finite Fields, Kluwer Academic Publishers, p. 25.

    Google Scholar 

  3. N. Courtois, Les cryptosystémes asymétriques à représentation obscure, Rapport de stage de DEA, Bull PTS — Université Paris 6, 1997. (This paper is available from J. Patarin and L. Goubin).

    Google Scholar 

  4. N. Demytko, A New Elliptic Curve Based Analogue of RSA, Advances in Cryptology, Proceedings of EUROCRYPT'93, Springer-Verlag, pp. 40–49.

    Google Scholar 

  5. M. Dickerson, The functional Decomposition of Polynomials, Ph.D Thesis, TR 89-1023, Department of Computer Science, Cornell University, Ithaca, NY, July 1989.

    Google Scholar 

  6. M. Garey, D. Johnson, Computers and Intractability, a Guide to the Theory of NP-Completeness, Freeman, p. 251.

    Google Scholar 

  7. K. Koyama, U.M. Maurer, T. Okamoto, S.A. Vanstone, New Public-Key Schemes Based on Elliptic Curves over the Ring Z n , Advances in Cryptology, Proceedings of CRYPTO'91, Springer-Verlag, pp. 252–266.

    Google Scholar 

  8. J.H. Loxton, D.S.P. Khoo, G.J. Bird, J. Seberry, A Cubic RSA Code Equivalent to Factorization, Journal of Cryptology, v.5, n.2, 1992, pp. 139–150.

    Article  Google Scholar 

  9. R. Lidl, G.L. Mullen, G. Turwald, Pitman Monographs and Surveys in Pure and Applied Mathematics 65: Dickson Polynomials, London, Longman Scientific and Technical, 1993.

    Google Scholar 

  10. R. Lidl, W.B. Müller, Permutation Polynomials in RSA-Cryptosystems, Advances in Cryptology, Proceedings of CRYPTO'83, Plenum Press, 1984, pp. 293–301.

    Google Scholar 

  11. T. Matsumoto, H. Imai, Algebraic Methods for Constructing Asymmetric Cryptosystems, AAECC-3, Grenoble, 1985.

    Google Scholar 

  12. T. Matsumoto, H. Imai, Public Quadratic Polynomial-Tuples for Efficient Signature-Verification and Message-Encryption, Advances in Cryptology, Proceedings of EUROCRYPT'88, Springer-Verlag, pp. 419–453.

    Google Scholar 

  13. W.B. Müller, Polynomial Functions in Modern Cryptology, Contributions to General Algebra 3: Proceedings of the Vienna Conference, Vienna: Verlag Holder-Pichler-Tempsky, 1985, pp. 7–32.

    Google Scholar 

  14. W.B. Müller, R. Nöbauer, Some Remarks on Public-Key Cryptography, Studia Scientiarum Mathematicarum Hungarica, v.16, 1981, pp. 71–76.

    Google Scholar 

  15. W.B. Müller, R. Nöbauer, Cryptanalysis of the Dickson-scheme, Advances in Cryptology, Proceedings of EUROCRYPT'85, Springer-Verlag, pp. 50–61.

    Google Scholar 

  16. J. Patarin, Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt'88, Advances in Cryptology, Proceedings of CRYPTO'95, Springer-Verlag, pp. 248–261.

    Google Scholar 

  17. J. Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP) Two New Families of Asymmetric Algorithms, Advances in Cryptology, Proceedings of EUROCRYPT'96, Springer-Verlag, pp. 33–48.

    Google Scholar 

  18. J. Patarin, Asymmetric Cryptography with a Hidden Monomial, Advances in Cryptology, Proceedings of CRYPTO'96, Springer-Verlag, pp. 45–60.

    Google Scholar 

  19. J. Patarin, L. Goubin, Asymmetric Cryptography with S-boxes, ICICS'97 (this conference.

    Google Scholar 

  20. M.O. Rabin, Digitized Signatures and Public-Key Functions as Intractable as Factorization, Technical Report LCS/TR-212, M.I.T. Laboratory for Computer Science, 1979.

    Google Scholar 

  21. R.L. Rivest, A. Shamir, L.M. Adleman, A Method for Obtaining Digital Signatures and Public-Key Cryptosystems, Communications of the ACM, v.21, n.2, 1978, pp. 120–126.

    Article  Google Scholar 

  22. P.J. Smith, LUC Public-Key Encryption, Dr. Dobb's Journal, January 1993, pp. 44–49.

    Google Scholar 

  23. P.J. Smith, M.J.J. Lennon, LUC. a New Public Key System, Proceedings of the Ninth IFIP Int. Symp. on Computer Security, 1993, pp. 103–117.

    Google Scholar 

  24. H.C. Williams, A Modification of the RSA Public-Key Encryption Procedure, IEEE Transactions on Information Theory, v.IT-26, n.6, 1980, pp. 726–729.

    Article  Google Scholar 

  25. H.C. Williams, An M 3 Public-Key Encryption Scheme, Advances in Cryptology, Proceedings of CRYPTO'85, Springer-Verlag, pp. 358–368.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bull PTS, 68 route de Versailles — BP 45, 78431, Louveciennes Cedex, France

    Jacques Patarin & Louis Goubin

Authors
  1. Jacques Patarin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Louis Goubin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Yongfei Han Tatsuaki Okamoto Sihan Qing

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag

About this paper

Cite this paper

Patarin, J., Goubin, L. (1997). Trapdoor one-way permutations and multivariate polynomials. In: Han, Y., Okamoto, T., Qing, S. (eds) Information and Communications Security. ICICS 1997. Lecture Notes in Computer Science, vol 1334. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028491

Download citation

  • DOI: https://doi.org/10.1007/BFb0028491

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63696-0

  • Online ISBN: 978-3-540-69628-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation