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-ladder for point multiplication. This Montgomery-halving can be run in parallel with the original Montgomery-ladder in order to concurrently compute part of the scalar multiplication. We also present two point thirding formulas in some subfamilies of curves $E(\mathbb {F}_{3^m})$ . We use these thirding formulas to implement scalar multiplication through (Third, Double)-and-add and (Third, Triple)-and-add parallel approaches. We also provide some implementation results of the presented parallel strategies which show a speed-up of 5-14 percent on an Intel Core i7 processor and a speed-up of 8-19 percent on a Qualcomm Snapdragon processor compared to non-parallelized approaches.
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