Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
Reducing elliptic curve logarithms to logarithms in a finite field
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
A new polynomial factorization algorithm and its implementation
Journal of Symbolic Computation
Elliptic Curve Public Key Cryptosystems
Elliptic Curve Public Key Cryptosystems
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
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
Schoof's algorithm and isogeny cycles
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Finding good random elliptic curves for cryptosystems defined over IF2n
EUROCRYPT'97 Proceedings of the 16th 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
Fail-Stop Threshold Signature Schemes Based on Elliptic Curves
ACISP '99 Proceedings of the 4th Australasian Conference on Information Security and Privacy
Finding Secure Curves with the Satoh-FGH Algorithm and an Early-Abort Strategy
EUROCRYPT '01 Proceedings of the International Conference on the Theory and Application of Cryptographic Techniques: Advances in Cryptology
Algebraic curves and cryptography
Finite Fields and Their Applications
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Schoof's algorithm is used to find a secure elliptic curve for cryptosystems, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as Atkin-Elkies's method, the isogeny cycles method, and trial search by match-and-sort techniques, we can count the number of rational points on an elliptic curve over GF(p) in a reasonable time, where p is a prime whose size is around 240-bits.