Advertisement

A Number-Theoretic Error-Correcting Code

  • Eric Brier
  • Jean-Sébastien Coron
  • Rémi Géraud
  • Diana MaimuţEmail author
  • David Naccache
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9522)

Abstract

In this paper we describe a new error-correcting code (ECC) inspired by the Naccache-Stern cryptosystem. While by far less efficient than Turbo codes, the proposed ECC happens to be more efficient than some established ECCs for certain sets of parameters.

The new ECC adds an appendix to the message. The appendix is the modular product of small primes representing the message bits. The receiver recomputes the product and detects transmission errors using modular division and lattice reduction.

Keywords

Expansion Rate Turbo Code Convolutional Code Message Size Noisy Channel 
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.

References

  1. 1.
    Berrou, C., Glavieux, A., Thitimajshima, P.: Near Shannon limit error-correcting coding and decoding: turbo-codes. In: IEEE International Conference on Communications - ICC 1993, vol. 2, pp. 1064–1070, May 1993Google Scholar
  2. 2.
    Chevallier-Mames, B., Naccache, D., Stern, J.: Linear bandwidth naccache-stern encryption. In: Ostrovsky, R., De Prisco, R., Visconti, I. (eds.) SCN 2008. LNCS, vol. 5229, pp. 327–339. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  3. 3.
    Dusart, P.: The \(k{\rm {th}}\) prime is greater than \(k (\ln k+ \ln \ln k-1)\) for \(k \ge 2\). Math. Comput. 68, 411–415 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Elias, P.: Coding for noisy channels. In: IRE Convention Record, pp. 37–46 (1955)Google Scholar
  5. 5.
    Fouque, P.-A., Stern, J., Wackers, G.-J.: CryptoComputing with rationals. In: Blaze, M. (ed.) FC 2002. LNCS, vol. 2357, pp. 136–146. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  6. 6.
    Goppa, V.D.: Codes on algebraic curves. Sov. Math. Dokl. 24, 170–172 (1981)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hamming, R.W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 29(2), 147–160 (1950)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Muller, D.E.: Application of boolean algebra to switching circuit design and to error detection. IRE Trans. Inf. Theory 3, 6–12 (1954)Google Scholar
  9. 9.
    Naccache, D., Stern, J.: A new public-key cryptosystem. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 27–36. Springer, Heidelberg (1997) Google Scholar
  10. 10.
    Reed, I.: A class of multiple-error-correcting codes and the decoding scheme. IRE Trans. Inf. Theory 4, 38–49 (1954)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Reed, I.S., Solomon, G.: Polynomial codes over certain finite fields. J. Soc. Ind. Appl. Math. 8(2), 300–304 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rosser, J.B.: The \(n\)-th prime is greater than \(n \ln n\). Proc. Lond. Math. Soc. 45, 21–44 (1938)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Shannon, C.: A mathematical theory of communication. Bell Syst. Tech. J. 27(379–423), 623–656 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vallée, B.: Gauss’ algorithm revisited. J. Algorithms 12(4), 556–572 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Eric Brier
    • 1
  • Jean-Sébastien Coron
    • 2
  • Rémi Géraud
    • 1
    • 3
  • Diana Maimuţ
    • 3
    Email author
  • David Naccache
    • 2
    • 3
  1. 1.IngenicoParisFrance
  2. 2.Université du LuxembourgLuxembourgLuxembourg
  3. 3.Département d’InformatiqueÉcole normale supérieureParis Cedex 05France

Personalised recommendations