On the group orders of elliptic curves over finite fields
By Everett W. Howe. 2001
53 citations recorded by the source corpus.
Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with F_q-defined points P and Q of order m and n, respectively, with P and (n/m)Q having a given Weil pairing. Using these curves, we estimate the number of elliptic curves over F_q that have a given integer N dividing the number of their F_q-defined points.
Subfields: Artificial Intelligence, Geometry and Topology, Information Systems
View Paper: https://www.semanticscholar.org/paper/676f10ae2a22e918866e3cbaac1eb07f9918d59a
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In Artificial Intelligence
In Geometry and Topology
In Information Systems
Citations in the corpus
- The Probability that the Number of Points on an Elliptic Curve over a Finite Field is Prime (2000, unknown): “Then, asymptotically, Pk 6= m Yi=1 1 p i i 1 p i+1 1 Y l pp; l62fp1 ;::: ;pmg 1 1l : For the analogous non-equality for elliptic curves, we need Howe's extension of Lenstra's probabilities to cover all small divisors of the number of points (see [3]): the probability that the number of points is div”
- Guide to Elliptic Curve Cryptography (2004, background): “Howe [201] extended Lenstra’s results to obtain, for any finite field Fq and prime power lk , estimates for the probability that a randomly selected elliptic curve over Fq has order #E(Fq ) divisible by lk .”