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Euclidean addition chains scalar multiplication on curves with efficient endomorphism

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Abstract

Random Euclidean addition chain generation has proven to be an efficient low memory and SPA secure alternative to standard ECC scalar multiplication methods in the context of fixed base point (Herbaut et al. in Progress in Cryptology-INDOCRYPT 2010, volume 6498 of LNCS. Springer, Berlin, pp 238–261, 2010). In this work, we show how to generalize this method to random point scalar multiplication on elliptic curves with an efficiently computable endomorphism. In order to do so, we generalize results from [21] on the relation of random Euclidean chains generation and elliptic curve point distribution obtained from those chains. We propose a software implementation of our method on various platforms to illustrate the impact of our approach. For that matter, we provide a comprehensive study of the practical computational cost of the modular multiplication when using Java and C standard libraries developed for the arithmetic over large integers.

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Acknowledgements

We would like to thank the referees for their careful reading and their helpful comments.

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Toulon, Université de Toulon, Toulon, France

    Yssouf Dosso, Fabien Herbaut, Nicolas Méloni & Pascal Véron

  2. Université de Nice Sophia Antipolis, Nice, France

    Fabien Herbaut

  3. ESPE Nice-Toulon, Nice, France

    Fabien Herbaut

Authors
  1. Yssouf Dosso
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  2. Fabien Herbaut
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  3. Nicolas Méloni
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  4. Pascal Véron
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Corresponding author

Correspondence to Pascal Véron.

Appendices

Appendix A: Benchmark platforms

All our source codes and collected results are available on GitHub: https://github.com/eacElliptic.

The characteristics of the platforms we used for our benchmarks are the following:

  • Android platform : Wiko Cik Peax 2 phone, with MediaTek MT6589 CPU (4 -core ARM Cortex-A7, 1.21 GHz), Android Version : 4.1.2, API Level 16.

  • Java platform : Intel Core I5-4210U 4-core 1.7Ghz, Broadwell technology, JDK 1.7.0_79, Ubuntu 14.04 LTS.

  • C platform : Intel Core I5-4210U 4-core 1.7Ghz, Broadwell technology, gcc 5.2.1, gmp 6.1.0, Ubuntu 14.04 LTS.

From various benchmarks, when the CPU models are exactly the same, it seems that on a given task Broadwell technology is better than Haswell technology by 5 percent to 10 percent or so. The Intel Turbo Boost technology has been disabled on the x64 platform so that the frequency of the processor be constant.

To collect the various execution results we have used the following tools:

  • Android platform: the startMethodTracing and the stopMethodTracing of the Debug class to generate trace logs, and the System.currentTimeMillis() method to measure execution time,

  • Java platform: the profiler provided with Netbeans IDE (v. 8.1) and the System.currentTimeMillis() method,

  • C platform: the clock.gettime() system call with CLOCK_PROCESS_CPUTIME_ID option, and taskset to bind our running process to only one processor.

Appendix B: Algorithms

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Appendix C: Anatomy of a modular multiplication

See Figs. 3 and 4

Fig. 3
figure 3

Anatomy of a C function computing a modular multiplication over 256-bit integers using Gnu MP (obtained from gprof)

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Fig. 4
figure 4

Anatomy of a Java method computing a modular multiplication over 256-bit integers using Big Integer Java library on an x64 platform (obtained from Netbeans profiler)

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Dosso, Y., Herbaut, F., Méloni, N. et al. Euclidean addition chains scalar multiplication on curves with efficient endomorphism. J Cryptogr Eng 8, 351–367 (2018). https://doi.org/10.1007/s13389-018-0190-0

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