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THE PROBABILITY THAT THE NUMBER OF POINTS ON AN ELLIPTIC CURVE OVER A FINITE FIELD IS PRIME

Published online by Cambridge University Press:  13 February 2001

STEVEN D. GALBRAITH  and
JAMES McKEE
STEVEN D. GALBRAITH
Affiliation:
Royal Holloway, University of London, Egham, Surrey TW20 0EX; [email protected]
JAMES McKEE
Affiliation:
Pembroke College, Oxford OX1 1DW; [email protected]
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Abstract

The paper gives a formula for the probability that a randomly chosen elliptic curve over a finite field has a prime number of points. Two heuristic arguments in support of the formula are given as well as experimental evidence. The paper also gives a formula for the probability that a randomly chosen elliptic curve over a finite field has kq points where k is a small number and q is a prime.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 62 , Issue 3 , December 2000 , pp. 671 - 684
Copyright
The London Mathematical Society 2000

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