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Finding the Eigenvalue in Elkies' Algorithm Müller, Volker ; 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)Counting the Number of Points on Elliptic Curves over Finite Fields of Characteristic Greater than Three ; ; Müller, Volker 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) |
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