Skip to main content

Advertisement

SpringerLink
Log in

Towards Minimal Addition Chains Using Ant Colony Optimisation

  • Published:
Journal of Mathematical Modelling and Algorithms

Abstract

Modular exponentiation is one of the most important operations in almost all modern cryptosystems. It is performed using a series of modular multiplications. This operation is time consuming for large operands as is always the case in cryptography. Hence fast public-key cryptography software or hardware requires optimisation of the time consumed by a single modular multiplication and/or the reduction of the total number of modular multiplications required. This paper introduces a novel idea based on the principles of ant colony optimisation for finding a minimal addition chain that allows one to reduce the number of modular multiplications so that modular exponentiation can be implemented efficiently. The best addition chain reached by the ant system is compared to the one used in the m-ary and sliding window methods as well as with the best addition chain evolved by genetic algorithms. We demonstrate that the ant system significantly outperforms all these methods for any exponent size.

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. Begeron, R., Berstel, J., Brlek, S. and Duboc, C.: Addition chains using continued fractions, J. Algorithms 10 (1989), 403–412.

    Article  MathSciNet  Google Scholar 

  2. Davis, L.: Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York, 1991.

    Google Scholar 

  3. Diaconis, P. and Efron, B.: Testing for independence in a two-way table: new interpretations of the chi-square statistic (with discussion), Ann. Stat. 13 (1985), 845–913.

    Article  MATH  MathSciNet  Google Scholar 

  4. Dorigo, M. and Gambardella, L. M.: Ant colony: a cooperative learning approach to the travelling salesman problem, IEEE Trans. Evol. Comput. 1(1) (1997), 53–66.

    Article  Google Scholar 

  5. Dorigo, M. and Stutzle, T.: Ant Colony Optimisation, MIT Press, 2004.

  6. Downing, P., Leong, B. and Sthi, R.: Computing sequences with addition chains, SIAM J. Comput. 10(3) (1981), 638–646.

    Article  MathSciNet  Google Scholar 

  7. Feber, J.: Multi-Agent Systems: an Introduction to Distributed Artificial Intelligence, Addison-Wesley, 1995.

  8. Haupt, R. L. and Haupt, S. E.: Practical Genetic Algorithms, Wiley, 1998.

  9. Knuth, D. E.: The Art of Programming: Semi-numerical Algorithms, Vol. 2, 2nd edn, Addison-Wesley, Reading, MA, 1981.

    Google Scholar 

  10. Koç, Ç. K.: High-speed RSA Implementation, Technical report, RSA Laboratories, Redwood City, California, USA, November 1994.

  11. Kunihiro, N. and Yamamoto, H.: New methods for generating short addition chain, IEICE Trans. E83-A(1) (January 2000), 60–67.

    Google Scholar 

  12. Menezes, A. J., Van Oorschot, P. C. and Vanstone, S. A.: Handbook of Applied Cryptography, CRC, 1996.

  13. Nedjah, N. and Mourelle, L. M.: Efficient parallel modular exponentiation algorithm, in Second International Conference on Information Systems, Lecture Notes in Computer Science, Vol. 2457, Springer, 2002, pp. 405–414.

  14. Nedjah, N. and Mourelle, L. M.: Minimal addition chains using genetic algorithms, in Proceedings of the Fifteenth International Conference on Industrial & Engineering Applications of Artificial Intelligence & Expert Systems, Lecture Notes in Computer Science, Vol. 2358, Springer, 2002, pp. 88–98.

  15. Nedjah, N. and Mourelle, L. M.: Efficient pre-processing for large window-based modular exponentiation using genetic algorithms, in Proceedings of the Sixteenth International Conference on Industrial & Engineering Applications of Artificial Intelligence & Expert Systems, Lecture Notes in Computer Science, Vol. 2718, Springer, 2003, pp. 625–635.

  16. Rivest, R., Shamir, A. and Adleman, L.: A method for obtaining digital signature and public-key cryptosystems, Commun. ACM (21) (1978), 120–126.

  17. Stutzle, T. and Dorigo, M.: ACO Algorithms for the Travelling Salesman Problems, Evolutionary Algorithms in Engineering and Computer Science, Wiley, 1999.

Download references

Author information

Authors and Affiliations

  1. Department of Electronics Engineering and Telecommunications, Faculty of Engineering, State University of Rio de Janeiro, Rua São Francisco Xavier, 524, Maracanã, Rio de Janeiro, RJ, Brazil

    Nadia Nedjah

  2. Department of Systems Engineering and Computation, Faculty of Engineering, State University of Rio de Janeiro, Rua São Francisco Xavier, 524, Maracanã, Rio de Janeiro, RJ, Brazil

    Luiza de Macedo Mourelle

Authors
  1. Nadia Nedjah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luiza de Macedo Mourelle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nadia Nedjah.

Additional information

★★ Research supported by FAPERJ (http://www.faperj.br) and CNPq (http://www.cnpq.br).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nedjah, N., de Macedo Mourelle, L. Towards Minimal Addition Chains Using Ant Colony Optimisation. J Math Model Algor 5, 525–543 (2006). https://doi.org/10.1007/s10852-005-9024-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10852-005-9024-z

Mathematical Subject Classifications (2000)

Key words

Search

Navigation