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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 Montgomeryhalving 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(F 3 m ). We use these thirdingdoi:10.1109/tc.2015.2389817 fatcat:5oa2quu4vzfofixv5h4wx5njqm