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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. The authors 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 that 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.
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