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Efficient CM-constructions of elliptic curves over finite fields. (English) Zbl 1127.14022
The authors present an algorithm that, on input an integer $$N\geq 1$$ together with its prime factorization, constructs a prime field $$\mathbb{F}_p$$ and an elliptic curve $$E$$ over $$\mathbb{F}_p$$ for which $$E(\mathbb{F}_p)$$ has order $$N$$. The algorithm uses the typical CM method, but chooses a prime $$p$$ leading to a relatively small class polynomial. The key idea is to find a small square-free integer $$d\geq1$$ together with an algebraic integer $$\alpha\in K=\mathbb{Q}(\sqrt{-d})$$ such that $$\text{N}_{K/\mathbb{Q}}(\alpha)=N$$ and $$\text{N}_{K/\mathbb{Q}}(1-\alpha)=p$$ is a prime.
There is no proof that such a pair $$(d,\alpha)$$ always exists, but under reasonable heuristic assumptions the run time is polynomial in $$2^{\omega(N)}\log N$$, where $$\omega(N)$$ is the number of prime factors of $$N$$. In the cryptographically relevant case where $$N$$ is prime an expected run time $$O((\log N)^{4+\varepsilon})$$ can be achieved.

##### MSC:
 14G15 Finite ground fields in algebraic geometry 11G15 Complex multiplication and moduli of abelian varieties 11G20 Curves over finite and local fields 94A60 Cryptography 14H52 Elliptic curves
ECPP
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