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Elliptic curves over finite fields with fixed subgroups

The order and group structure of an elliptic curve over a finite field is of great theoretical and practical interest. We will focus on a practical application, specif- ically on factoring using elliptic curves. The elliptic curve factoring method was discovered by Lenstra [6] in 1987 and is still the best algorithm for finding medium sized factors of a composite number. The choice of the elliptic curve for the factoring method is important. In general, one hopes that E(Fp), where p is a prime factor we want to find, will be smooth. Atkin and Morain [1] suggested using elliptic curves with large rational torsion, because the torsion subgroup injects into E(Fp) for all except a few p. This makes the order of the elliptic curve divisible by the order of the torsion, and

elliptic curves

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