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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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