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On Elkies subgroups of $$\ell$$-torsion points in elliptic curves defined over a finite field. (English) Zbl 1230.11080
Summary: As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic $$p$$, there exists an algorithm that computes, for $$\ell$$ an Elkies prime, $$\ell$$-torsion points in an extension of degree $$\ell-1$$ at cost $$\tilde O(\ell\max(\ell,\log q)^2)$$ bit operations in the favorable case where $$\ell\leq p/2$$.
We combine in this work a fast algorithm for computing isogenies due to A. Bostan, F. Morain, B. Salvy and É. Schost [Math. Comput. 77, No. 263, 1755–1778 (2008; Zbl 1200.11097)] with the $$p$$-adic approach followed by Joux and Lercier to get an algorithm valid without any limitation on $$\ell$$ and $$p$$ but of similar complexity. For the sake of simplicity, we precisely state here the algorithm in the case of finite fields with characteristic $$p\geq 5$$. We give experiment results, too.

##### MSC:
 11G20 Curves over finite and local fields 11Y16 Number-theoretic algorithms; complexity
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##### References:
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