Efficient CM-constructions of elliptic curves over finite fields

Abstract

We present an algorithm that, on input of an integer N ≥ 1 together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) has order N. Although it is unproved that this can be done for all N , a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2 ω(N) log N , where ω(N) is the number of distinct prime factors of N. In the cryptographically relevant case where N is prime, an expected run time O((log N) 4+ε) can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order N = 10 2004 and N = nextprime(10 2004) = 10 2004 +4863.