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Speeding scalar multiplication over binary elliptic curves using the new carry-less multiplication instruction

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Abstract

The availability of a new carry-less multiplication instruction in the latest Intel desktop processors significantly accelerates multiplication in binary fields and hence presents the opportunity for reevaluating algorithms for binary field arithmetic and scalar multiplication over elliptic curves. We describe how to best employ this instruction in field multiplication and the effect on performance of doubling and halving operations. Alternate strategies for implementing inversion and half-trace are examined to restore most of their competitiveness relative to the new multiplier. These improvements in field arithmetic are complemented by a study on serial and parallel approaches for Koblitz and random curves, where parallelization strategies are implemented and compared. The contributions are illustrated with experimental results improving the state-of-the-art performance of halving and doubling-based scalar multiplication on NIST curves at the 112- and 192-bit security levels and a new speed record for side-channel-resistant scalar multiplication in a random curve at the 128-bit security level. The algorithms presented in this work were implemented on Westmere and Sandy Bridge processors, the latest generation Intel microarchitectures.

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Author information

Authors and Affiliations

  1. Université de Lyon, Université Lyon 1, ISFA, Lyon, France

    Jonathan Taverne

  2. Computer Science Department, CINVESTAV-IPN, Mexico City, Mexico

    Armando Faz-Hernández & Francisco Rodríguez-Henríquez

  3. Institute of Computing, University of Campinas, Campinas, Brazil

    Diego F. Aranha & Julio López

  4. Auburn University, Auburn, USA

    Darrel Hankerson

Authors
  1. Jonathan Taverne

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  2. Armando Faz-Hernández

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  3. Diego F. Aranha

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  4. Francisco Rodríguez-Henríquez

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  5. Darrel Hankerson

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  6. Julio López

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Corresponding author

Correspondence toFrancisco Rodríguez-Henríquez.

Additional information

J. Taverne: This work was performed while the author was visiting CINVESTAV-IPN. D. F. Aranha: A portion of this work was performed while the author was visiting University of Waterloo.

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Taverne, J., Faz-Hernández, A., Aranha, D.F.et al. Speeding scalar multiplication over binary elliptic curves using the new carry-less multiplication instruction.J Cryptogr Eng1, 187–199 (2011). https://doi.org/10.1007/s13389-011-0017-8

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