New Parallel Approaches for Scalar Multiplication in Elliptic Curve over Fields of Small Characteristic

Christophe Negre 1 Jean-Marc Robert 1
1 DALI - Digits, Architectures et Logiciels Informatiques
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier, UPVD - Université de Perpignan Via Domitia
Abstract : We present two new strategies for parallel implementation of scalar multiplication over elliptic curves. We first introduce a Montgomery-halving algorithm which is a variation of the original Montgomery point multiplication. This Montgomery-halving can be run in parallel with the original Montgomery point multiplication in order to concurently compute part of the scalar multiplication. We also present two point thirding formulas in some subfamily of curves E(GF(3^m)). We use these thirding formulas to implement the scalar multiplication through a Third-and-add approach and a parallel Third-and-add and Double-and-add or Triple-and-add approaches. We also provide some implementation results on an Intel Core i7 of the presented two strategies which show a speed-up of 5%-13% compared to non-parallelized approaches.
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Christophe Negre, Jean-Marc Robert. New Parallel Approaches for Scalar Multiplication in Elliptic Curve over Fields of Small Characteristic. IEEE Transactions on Computers, Institute of Electrical and Electronics Engineers, 2015, 64 (10), pp.2875-2890. ⟨http://doi.ieeecomputersociety.org⟩. ⟨10.1109/TC.2015.2389817⟩. ⟨hal-00908463⟩

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