Finding elliptic curves with a subgroup of prescribed size

Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m = o ( p 1 / 2 ( log p ) − 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 m p 1 / 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.