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Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m = o(p1/2(logp)−4), outputs an elliptic curve E over the finite field… Click to show full abstract
Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m = o(p1/2(logp)−4), outputs an elliptic curve E over the finite field ????p for which the cardinality of E(????p) is divisible by m. The running time of the algorithm is mp1/2+o(1), and this leads to more efficient constructions of rational functions over ????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.
Journal Title: International Journal of Number Theory
Year Published: 2017
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