Elliptic curves over finite fields with fixed subgroups
نویسنده
چکیده
We prove that for any given group Zm⊕Zn, where m divides n, and any rational elliptic curve, for a positive density of the rational primes p ∈ P, Zm ⊕ Zn is isomorphic to a subgroup of E(Fp). Our methods are effective and we demonstrate how to construct elliptic curves such that for a large density of the primes p, the given group is isomorphic to a subgroup of E(Fp). We show that for some groups G, one can use elliptic curves over number fields and reduce them to elliptic curves over finite fields having G as a subgroup for a large density of the fields. We also discuss heuristics how to choose good elliptic curves for integer factorization with elliptic curves.
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