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This paper presents a series of Montgomery scalar multiplication algorithms on general short Weierstrass curves over fields with characteristic greater than 3, which need only 12 field multiplications per scalar bit using 8 \begin{document}$ \sim $\end{document} 9 field registers, thus outperform the binary NAF method on average. Over binary fields, the Montgomery scalar multiplication algorithm which was presented at the first CHES workshop by Lopez and Dahab has been a favorite of ECC implementors, due to its nice properties such as high efficiency (outperforming the binary NAF), natural SPA-resistance, generality (coping with all ordinary curves) and implementation easiness. Over odd characteristic fields, the new scalar multiplication algorithms are the first ones featuring all these properties. Building-blocks of our contribution are new efficient differential addition-and-doubling formulae and a novel conception of on-the-fly adaptive coordinates which varies in accordance with not only the base point but also the bits of the given scalar.

Speeding up regular elliptic curve scalar multiplication without precomputation