# Explicit isogenies in quadratic time in any characteristic

3 Computer Science Department, University of Western Ontario
ORCCA - Ontario Research Center for Computer Algebra, Computer Science Department
Abstract : Consider two ordinary elliptic curves $E,E'$ defined over a finite field $\F_q$, and suppose that there exists an isogeny $\psi$ between $E$ and $E'$. We propose an algorithm that determines $\psi$ from the knowledge of $E$, $E'$ and of its degree $r$, by using the structure of the $ℓ$-torsion of the curves (where $ℓ$~is a prime different from the characteristic~$p$ of the base field). Our approach is inspired by a previous algorithm due to Couveignes, that involved computations using the $p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of~$\tildO(r^2) p^{O(1)}$ base field operations. On the other hand, the cost of our algorithm is $\tildO(r^2) \log(q)^{O(1)}$, for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.
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https://hal.archives-ouvertes.fr/hal-01320940
Contributor : Cyril Hugounenq <>
Submitted on : Friday, May 27, 2016 - 4:43:25 PM
Last modification on : Thursday, February 7, 2019 - 4:47:16 PM
Long-term archiving on : Sunday, August 28, 2016 - 10:18:25 AM

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explicit-isogenies.pdf
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### Identifiers

• HAL Id : hal-01320940, version 1
• ARXIV : 1603.00711

### Citation

Luca de Feo, Cyril Hugounenq, Jérôme Plût, Eric Schost. Explicit isogenies in quadratic time in any characteristic. 2016. ⟨hal-01320940⟩

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