Abstract
A variation of the Complex Multiplication (CM) method for generating elliptic curves of known order over finite fields is proposed. We give heuristics and timing statistics in the mildly restricted setting of prime curve order. These may be seen to corroborate earlier work of Koblitz in the class number one setting. Our heuristics are based upon a recent conjecture by R. Gross and J. Smith on numbers of twin primes in algebraic number fields.
Our variation precalculates class polynomials as a separate off-line process. Unlike the standard approach, which begins with a prime p and searches for an appropriate discriminant D, we choose a discriminant and then search for appropriate primes. Our on-line process is quick and can be compactly coded.
In practice, elliptic curves with near prime order are used. Thus, our timing estimates and data can be regarded as upper estimates for practical purposes.
This research was supported by rTrust Technologies.
The reader should note that Oregon State University has filed US and International patent applications for inventions described in this paper.
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Savaş, E., Schmidt, T.A., Koç, Ç.K. (2001). Generating Elliptic Curves of Prime Order. In: Koç, Ç.K., Naccache, D., Paar, C. (eds) Cryptographic Hardware and Embedded Systems — CHES 2001. CHES 2001. Lecture Notes in Computer Science, vol 2162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44709-1_13
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