Finding elliptic curves with a subgroup of prescribed size
Number Theory
2017-01-03 v3
Abstract
Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m=o(sqrt(p)/(log p)^4), outputs an elliptic curve E over the finite field F_p for which the cardinality of E(F_p) is divisible by m. The running time of the algorithm is mp^(1/2+o(1)), and this leads to more efficient constructions of rational functions over F_p whose image is small relative to p. We also give an unconditional version of the algorithm that works for almost all primes p, and give a probabilistic algorithm with subexponential time complexity.
Keywords
Cite
@article{arxiv.1403.7887,
title = {Finding elliptic curves with a subgroup of prescribed size},
author = {Igor E. Shparlinski and Andrew V. Sutherland},
journal= {arXiv preprint arXiv:1403.7887},
year = {2017}
}
Comments
21 pages, minor corrections, added a new section