Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
A course in computational algebraic number theory
A course in computational algebraic number theory
Elliptic Curve Public Key Cryptosystems
Elliptic Curve Public Key Cryptosystems
Efficient construction of secure hyperelliptic discrete logarithm problems
ICICS '97 Proceedings of the First International Conference on Information and Communication Security
Design of Elliptic Curves with Controllable Lower Boundary of Extension Degree for Reduction Attacks
CRYPTO '94 Proceedings of the 14th Annual International Cryptology Conference on 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
Schoof's algorithm and isogeny cycles
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Building cyclic elliptic curves modulo large primes
EUROCRYPT'91 Proceedings of the 10th annual international conference on Theory and application of cryptographic techniques
Counting the number of points on elliptic curves over finite fields: strategies and performances
EUROCRYPT'95 Proceedings of the 14th annual international conference on Theory and application of cryptographic techniques
Characterization of Elliptic Curve Traces under FR-Reduction
ICISC '00 Proceedings of the Third International Conference on Information Security and Cryptology
Isogeny Volcanoes and the SEA Algorithm
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
Action of Modular Correspondences around CM Points
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
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Elliptic curves over number fields with CM can be used to design non-isogenous elliptic cryptosystems over finite fields efficiently. The existing algorithm to build such CM curves, so-called the CM field algorithm, is based on analytic expansion of modular functions, costing computations of O(25h/2h21/4) where h is the class number of the endomorphism ring of the CM curve. Thus it is effective only in the small class number cases. This paper presents polynomial time algorithms in h to build CM elliptic curves over number fields. In the first part, probabilistic probabilistic algorithms of CM tests are presented to find elliptic curves with CM without restriction on class numbers. In the second part, we show how to construct ring class fields from ray class fields. Finally, a deterministic algorithm for lifting the ring class equations from small finite fields thus construct CM curves is presented. Its complexity is shown as O(h7).