Skip to main content
SpringerLink
Log in

McEliece public key cryptosystems using algebraic-geometric codes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

McEliece proposed a public-key cryptosystem based on algebraic codes, in particular binary classical Goppa codes. Actually, his scheme needs only a class of codes with a good decoding algorithm and with a huge number of inequivalent members with given parameters. In the present paper we look at various aspects of McEliece's scheme using the new and much larger class of q-ary algebraic-geometric Goppa codes.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. M. Adams and H. Meijer, Security-related comments regarding McEliece public-key cryptosystems: Advances in Cryptology-CRYPTO '87, Springer-Verlag, New York (1987) pp. 224–228.

    Google Scholar 

  2. E. R. Berlekamp, R. J. McEliece and H. C. A. van Tilborg, On the inherent intractability of certain coding problems, IEEE Trans. Inform. Th., Vol. IT-24 (1981) pp. 384–386.

    Google Scholar 

  3. Th. Beth, M. Frisch and G. J. Simmons (Eds.), Public-Key Cryptography: State of the Art and Future Directions, Lecture Notes in Computer Science, Springer-Verlag, 578 (1992).

  4. D. E. Denning, Cryptography and Data Security, Addison-Wesley, Reading, MA (1982).

    Google Scholar 

  5. I. I. Dumer, V. A. Zinoviev and V. V. Zyablov, Concatenated decoding according to minimal generalized distance, Problems of Control and Information Theory, Vol. 10, No. 1 (1981) pp. 3–19.

    Google Scholar 

  6. I. M. Duursma, Algebraic decoding using special divisors, IEEE Transaction on Information Theory, Vol. 39, No. 2, pp. 694–698.

  7. I. M. Duursma, Majority coset decoding, IEEE Transaction on Information Theory, Vol. 39, No. 3 (May 1993) pp. 1067–1070.

    Google Scholar 

  8. Dirk Ehrhard, Achieving the designed error capacity in decoding algebraic-geometric codes, IEEE Transaction on Information Theory, Vol. 39, No. 3 (May 1993) pp. 743–751.

    Google Scholar 

  9. G. L. Feng and T. R. Rao, Decoding algebraic-geometric codes up to designed minimum distance, IEEE Trans. on Inform. Theory, Vol. 39 (1993) pp. 37–45.

    Google Scholar 

  10. G. L. Feng, V. K. Wei, T. R. Rao and K. K. Tzeng, Simplified understanding and efficient decoding of a class of algebraic-geometric codes, IEEE Trans. on Inform. Theory, Vol. 40, No. 4 (1994) pp. 981.

    Google Scholar 

  11. G. D. Forney, Concatenated Codes, MIT Press, Cambridge, MA (1966).

    Google Scholar 

  12. J. K. Gibbon, Equivalent Goppa codes and trapdoors to McEliece's public-key cryptosystem: EUROCRYPT '91, Lect. Notes in CS, 547 (1991) pp. 68–70.

  13. H. L. Janwa, l-MDS codes, threshold schemes and algebraic-geometric codes, submitted to IEEE Transactions on Information Theory.

  14. J. Justesen, K. L. Larsen, H. E. Jensen, A. Havemose, T. Høholdt, Construction and decoding of a class of algebraic-geometric codes, IEEE Trans. Inform. Th., Vol. IT-35 (1989) pp. 811–821.

    Google Scholar 

  15. J. Justesen, K. L. Larsen, H. E. Jensen, and T. Høholdt, Fast decoding of codes from algebraic-plane curves, IEEE Trans. Inform. Th, Vol. IT-38 (Jan 1992) pp. 111–119.

    Google Scholar 

  16. V. I. Korzhik and A. I. Turkin, Cryptanalysis of McEliece's public-key, cryptosystem: EUROCRYPT '91, Lect. Notes in CS, 547 (1991) pp. 68–70.

    Google Scholar 

  17. D. Le Brigand and J. J. Risler, Algorithms de Brill-Noether et codes de Goppa, Bull. Soc. Math. France, Vol. 116 (1988) pp. 231–253.

    Google Scholar 

  18. P. J. Lee and E. F. Brickell, An observation on the security of McEliece's public-key cryptosystem: Advances in Cryptology-EUROCRYPT '88, Springer LNCS, 330 (1988) pp. 275–280.

  19. R. J. McEliece, A Public-key cryptosystem based on algebraic coding theory, DSN Progress Report, Jet Propulsion Laboratory, Pasadena, CA (Jan./Feb. 1978) pp. 114–116.

    Google Scholar 

  20. C. Moreno, Algebraic curves over finite fields, Cambridge Tracts in Mathematics, Cambridge University Press, No. 97 (1991).

  21. H. Niederreiter, Knapsack-type cryptosystems and algebraic coding theory, Problems of Control and Information Theory, Vol. 15, No. 2 (1986) pp. 159–166.

    Google Scholar 

  22. C. S. Park, Improving code rate of McEliece's public-key cryptosystem, Electronics Letters, Vol. 25, No. 21 (1989) pp. 1466–1467.

    Google Scholar 

  23. N. Patterson, The algebraic decoding of Goppa codes, IEEE Trans. on Information Theory 21, (1975) pp. 203–207.

    Google Scholar 

  24. R. Pellikaan, On a decoding algorithm for codes on maximal curves, IEEE Trans. Inform. Th., Vol. IT-35, (1989) pp. 1228–1232.

    Google Scholar 

  25. D. Polemi, C. Moreno and O. Moreno, A construction of a.g. Goppa codes from singular curves, preprint.

  26. D. Polemi, M. Hasner, O. Moreno and C. Williamson, A Computer algebra algorithm for the adjoint divisor: Proc. of IEEE IT Symposium, San Antonio, Texas (1993) p. 358.

  27. S. C. Porter, B-Z. Shen and R. Pellikaan, Decoding geometric Goppa codes using an extra place, IEEE Trans. Inform. Th., Vol. 38 (Nov. 1992) pp. 1963–1976.

    Google Scholar 

  28. T. R. N. Rao and Kil-H. Nam, Private-key algebraic-code encryption, IEEE Trans. Inform. Th., Vol. IT-35 (1989).

  29. S. Sakata, J. Justesen, Y. Madelung, H. E. Jensen and T. Høholdt, A fast decoding method of AG codes from Miura-Kamiya curves C ab up to half the Reng-Rao bound. Finite Fields and Their Applications, Vol. 1, No. 1 (January, 1995) pp. 83–101.

    Google Scholar 

  30. Jean-Pierre Serre, Nombres de points des courbes Algébriques sur F q , Séminaire de Théorie des Nombres de Bordeaux, exposé 22 (1983) pp. 1–8.

  31. Jean-Pierre Serre, Rational points on curves over finite fields, “q Large”, Parts I and II, Lectures given at Harvard University, (September–December, 1985). Notes by Fernando Gouvea, Serre (1985).

  32. Gustavus J. Simmons (ed.), Contemporary Cryptology: The Science of Information Integrity, IEEE Press, New Jersey (1992).

    Google Scholar 

  33. A. N. Skorobogatov and S. G. 307–02, On the decoding of algebraic-geometric codes, IEEE Trans. Inform. Theory, Vol. 36, No. 5 (1990) pp. 1051–1060.

    Google Scholar 

  34. Y. Sugiyama et al., Further results on Goppa codes and their applications to constructing efficient binary codes, IEEE Trans. Inform. Theory, Vol. 22 (1976) pp. 518–526.

    Google Scholar 

  35. J. van Tilburg, On the McEliece public-key cryptosystem: CRYPTO '88, Lecture Notes in CS, 403 (1988).

  36. M. A. Tsfasman and S. G. Vlădut, Algebraic-geometric codes, Kluwer Akad. Publ. (1991).

  37. S. G. Vlàdut, On the decoding of algebraic-geometric codes over F q for q ≥ 16, IEEE Trans. Inform. Th., Vol. IT-36 (1990) pp. 1461–1463.

    Google Scholar 

  38. J. Wolfmann, The number of rational points on certain algebraic curves over finite fields, Communications in Algebra, Vol. 17, No. 8 (1989) pp. 2055–2060.

    Google Scholar 

  39. M. Wirtz, On the parameters of Goppa codes, IEEE Trans. Inform. Th., Vol. 34, No. 5 (Sept. 1988) pp. 1341–1343.

    Google Scholar 

  40. V. A. Zinoviev, Generalized concatenated codes for channels with bursts of errors and independent errors, Problems of Inform. Trans., Vol. 17, No. 4 (1981) pp. 53–62.

    Google Scholar 

  41. G. A. Kabatianskii, On security of McEliee and Niederreiter type cryptosystems, Lecture delivered at the University of Puerto Rico, (September 1993).

  42. E. Krouk, A new public key cryptosystem: Proceedings of the Sixth Swedish-Russian International Workshop on Information Theory, (1993) pp. 285–286.

  43. V. M. Sidelnikov and S. O. Shestakov, On insecurity of cryptosystems based on generalized Reed-Solomon codes, Diskretnaya Matematika, Vol. 4, No. 3 (1992). Translated in, Discrete Math. Appl., Vol. 2, No. 4 (1992) pp. 439–444.

Download references

Author information

Authors and Affiliations

  1. The Mehta Research Institute of Mathematics and Mathematical Physics, 211 002, Allahabad, India

    Heeralal Janwa

  2. Department of Mathematics, University of Puerto Rico, 00931, Rio Piedras, PR, Puerto Rico

    Oscar Moreno

Authors
  1. Heeralal Janwa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oscar Moreno
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by: S. A. Vanstone

Rights and permissions

Reprints and permissions

About this article

Cite this article

Janwa, H., Moreno, O. McEliece public key cryptosystems using algebraic-geometric codes. Des Codes Crypt 8, 293–307 (1996). https://doi.org/10.1007/BF00173300

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00173300

Keywords

Search

Navigation