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Evolutionary Algorithms for Finding Short Addition Chains: Going the Distance

  • Stjepan Picek
  • Carlos A. Coello Coello
  • Domagoj Jakobovic
  • Nele Mentens
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9595)

Abstract

The problem of finding the shortest addition chain for a given exponent is of great relevance in cryptography, but is also very difficult to solve since it is an NP-hard problem. In this paper, we propose a genetic algorithm with a novel representation of solutions and new crossover and mutation operators to minimize the length of the addition chains corresponding to a given exponent. We also develop a repair strategy that significantly enhances the performance of our approach. The results are compared with respect to those generated by other metaheuristics for instances of moderate size, but we also investigate values up to \(2^{127} - 3\). For those instances, we were unable to find any results produced by other metaheuristics for comparison, and three additional strategies were adopted in this case to serve as benchmarks. Our results indicate that the proposed approach is a very promising alternative to deal with this problem.

Keywords

Addition chains Cryptography Genetic algorithms Exponentiation 

Notes

Acknowledgments

This work has been supported in part by Croatian Science Foundation under the project IP-2014-09-4882. The second author gratefully acknowledges support from CONACyT project no. 221551. In addition, this work was supported in part by the Research Council KU Leuven (C16/15/058) and IOF project EDA-DSE (HB/13/020).

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Stjepan Picek
    • 1
  • Carlos A. Coello Coello
    • 2
  • Domagoj Jakobovic
    • 3
  • Nele Mentens
    • 1
  1. 1.ESAT/COSIC and iMindsKU LeuvenLeuven-HeverleeBelgium
  2. 2.Department of Computer ScienceCINVESTAV-IPNMexico D.F.Mexico
  3. 3.Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia

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