Abstract
We propose a method for increasing the speed of scalar multiplication on binary anomalous (Koblitz) elliptic curves. By introducing a generator which produces random pairs (k, [k]P) of special shape, we exhibit a specific setting where the number of elliptic curve operations is reduced by 25% to 50% compared with the general case when k is chosen uniformly. This generator can be used when an ephemeral pair (k, [k]P) is needed by a cryptographic algorithm, and especially for Elliptic Curve Diffie-Hellman key exchange, ECDSA signature and El-Gamal encryption. The presented algorithm combines normal and polynomial basis operations to achieve optimal performance. We prove that a probabilistic signature scheme using our generator remains secure against chosen message attacks.
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Coron, JS., M’Raïhi, D., Tymen, C. (2001). Fast Generation of Pairs (k, [k]P) for Koblitz Elliptic Curves. In: Vaudenay, S., Youssef, A.M. (eds) Selected Areas in Cryptography. SAC 2001. Lecture Notes in Computer Science, vol 2259. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45537-X_12
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