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Counting Points on Elliptic Curves Over Finite Fields
, 1995
"... . We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large; it is based on Shanks's babystepgiantstep strategy. The second algorithm is very efficient when the endomorphism ring of ..."
Abstract

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. We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large; it is based on Shanks's babystepgiantstep strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic polynomial time algorithm was impractical in its original form. We discuss several practical improvements by Atkin and Elkies. 1. Introduction. Let p be a large prime and let E be an elliptic curve over F p given by a Weierstraß equation Y 2 = X 3 +AX +B for some A, B 2 F p . Since the curve is not singular we have that 4A 3 + 27B 2 6j 0 (mod p). We describe several methods to count the rational points on E, i.e., methods to determine the number of points (x; y) on E with x; y 2 F p . Most o...
Algorithms for computing isogenies between elliptic curves
 Math. Comp
, 2000
"... Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes sh ..."
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Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes’s algorithm. In particular, we describe the use of fast algorithms for performing incremental operations on series. We also insist on the particular case of the characteristic 2. 1.
ECC: Do We Need to Count?
, 1999
"... A prohibitive barrier faced by elliptic curve users is the difficulty of computing the curves' cardinalities. Despite recent theoretical breakthroughs, point counting still remains very cumbersome and intensively time consuming. ..."
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A prohibitive barrier faced by elliptic curve users is the difficulty of computing the curves' cardinalities. Despite recent theoretical breakthroughs, point counting still remains very cumbersome and intensively time consuming.