A course in computational algebraic number theory
A course in computational algebraic number theory
Mathematics of Computation
The State of Elliptic Curve Cryptography
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
Efficient Identity Based Parameter Selection for Elliptic Curve Cryptosystems
ACISP '99 Proceedings of the 4th Australasian Conference on Information Security and Privacy
Elliptic Curves over Fp Suitable for Cryptosystems
ASIACRYPT '92 Proceedings of the Workshop on the Theory and Application of Cryptographic Techniques: Advances in Cryptology
Constructing elliptic curves with given group order over large finite fields
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
A Software Library for Elliptic Curve Cryptography
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
On the Efficient Generation of Elliptic Curves over Prime Fields
CHES '02 Revised Papers from the 4th International Workshop on Cryptographic Hardware and Embedded Systems
A method for distinguishing the two candidate elliptic curves in CM method
ICISC'04 Proceedings of the 7th international conference on Information Security and Cryptology
Generating prime order elliptic curves: difficulties and efficiency considerations
ICISC'04 Proceedings of the 7th international conference on Information Security and Cryptology
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Av ariation 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.