Counting points on elliptic curves over 𝐅_{2^{𝐦}}

Menezes, Alfred J.; Vanstone, Scott A.; Zuccherato, Robert J.

doi:10.1090/S0025-5718-1993-1153167-9

Counting points on elliptic curves over $\mathbf {F}_{2^m}$
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by Alfred J. Menezes, Scott A. Vanstone and Robert J. Zuccherato PDF

Math. Comp. 60 (1993), 407-420 Request permission

Abstract:

In this paper we present an implementation of Schoof’s algorithm for computing the number of ${F_{{2^m}}}$-points of an elliptic curve that is defined over the finite field ${F_{{2^m}}}$. We have implemented some heuristic improvements, and give running times for various problem instances.

References

Alfred Menezes and Scott Vanstone, The implementation of elliptic curve cryptosystems, Advances in cryptology—AUSCRYPT ’90 (Sydney, 1990) Lecture Notes in Comput. Sci., vol. 453, Springer, Berlin, 1990, pp. 2–13. MR 1083769, DOI 10.1525/awr.1990.11.1.2

The number of points on an elliptic curve modulo a prime

Computing the number of points of elliptic curves over finite fields

Enumeration of rational points on elliptic curves over finite fields

Neal Koblitz, Elliptic curve cryptosystems, Math. Comp. 48 (1987), no. 177, 203–209. MR 866109, DOI 10.1090/S0025-5718-1987-0866109-5
Neal Koblitz, Constructing elliptic curve cryptosystems in characteristic $2$, Advances in cryptology—CRYPTO ’90 (Santa Barbara, CA, 1990) Lecture Notes in Comput. Sci., vol. 537, Springer, Berlin, 1991, pp. 156–167. MR 1232869, DOI 10.1007/3-540-38424-3_{1}1
Neal Koblitz, CM-curves with good cryptographic properties, Advances in cryptology—CRYPTO ’91 (Santa Barbara, CA, 1991) Lecture Notes in Comput. Sci., vol. 576, Springer, Berlin, 1992, pp. 279–287. MR 1243654, DOI 10.1007/3-540-46766-1_{2}2
Serge Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin-New York, 1978. MR 518817, DOI 10.1007/978-3-662-07010-9
Alfred Menezes and Scott Vanstone, Isomorphism classes of elliptic curves over finite fields of characteristic $2$, Utilitas Math. 38 (1990), 135–153. MR 1093882
Victor S. Miller, Use of elliptic curves in cryptography, Advances in cryptology—CRYPTO ’85 (Santa Barbara, Calif., 1985) Lecture Notes in Comput. Sci., vol. 218, Springer, Berlin, 1986, pp. 417–426. MR 851432, DOI 10.1007/3-540-39799-X_{3}1
R. C. Mullin, I. M. Onyszchuk, S. A. Vanstone, and R. M. Wilson, Optimal normal bases in $\textrm {GF}(p^n)$, Discrete Appl. Math. 22 (1988/89), no. 2, 149–161. MR 978054, DOI 10.1016/0166-218X(88)90090-X
J. M. Pollard, Monte Carlo methods for index computation $(\textrm {mod}\ p)$, Math. Comp. 32 (1978), no. 143, 918–924. MR 491431, DOI 10.1090/S0025-5718-1978-0491431-9
René Schoof, Elliptic curves over finite fields and the computation of square roots mod $p$, Math. Comp. 44 (1985), no. 170, 483–494. MR 777280, DOI 10.1090/S0025-5718-1985-0777280-6