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See detailFinding the Eigenvalue in Elkies' Algorithm
Müller, Volker UL; Maurer, Markus

in Experimental Mathematics (2001), 10(2), 275-285

One important part of Elkies' algorithm for computing the group order of an elliptic curve is the search for an eigenvalue of the Frobenius endomorphism. In this paper we compare two well known algorithms ... [more ▼]

One important part of Elkies' algorithm for computing the group order of an elliptic curve is the search for an eigenvalue of the Frobenius endomorphism. In this paper we compare two well known algorithms with two new ideas based on the Babystep Giantstep method. Moreover we show how resultants can be used to speed up this search. Finally we present a fast probabilistic algorithm for checking whether a given rational function is congruent to an entry in a table of rational functions modulo some fixed polynomial. [less ▲]

Detailed reference viewed: 90 (3 UL)
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See detailCounting the Number of Points on Elliptic Curves over Finite Fields of Characteristic Greater than Three
Lehmann, Frank; Maurer, Markus; Müller, Volker UL et al

in Proceedings of Algorithmic Number Theory Symposium I, Lecture Notes in Computer Science (1994)

The paper describes the implementation of the Algorithm of Atkin and Elkies for computing the group order of elliptic curves over large prime fields. The focus of the paper lays on algorithmical aspects ... [more ▼]

The paper describes the implementation of the Algorithm of Atkin and Elkies for computing the group order of elliptic curves over large prime fields. The focus of the paper lays on algorithmical aspects of the Atkin/Elkies algorithm. Practical data and running times are given. [less ▲]

Detailed reference viewed: 82 (2 UL)