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Explicit isogenies in quadratic time in any characteristic

Part of: Curves Arithmetic algebraic geometry Computational number theory

Published online by Cambridge University Press:  26 August 2016

Luca De Feo Open the ORCID record for Luca De Feo [Opens in a new window] ,
Cyril Hugounenq ,
Jérôme Plût  and
Éric Schost
Luca De Feo
Affiliation:
LMV – UVSQ, 45 avenue des États-Unis, 78035 Versailles, France, email [email protected]
Cyril Hugounenq
Affiliation:
LMV – UVSQ, 45 avenue des États-Unis, 78035 Versailles, France email [email protected]
Jérôme Plût
Affiliation:
ANSSI, 51, boulevard de La Tour-Maubourg, 75007 Paris, France email [email protected]
Éric Schost
Affiliation:
Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada N2L 3G1 email [email protected]

Abstract

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Consider two ordinary elliptic curves $E,E^{\prime }$ defined over a finite field $\mathbb{F}_{q}$, and suppose that there exists an isogeny $\unicode[STIX]{x1D713}$ between $E$ and $E^{\prime }$. We propose an algorithm that determines $\unicode[STIX]{x1D713}$ from the knowledge of $E$, $E^{\prime }$ and of its degree $r$, by using the structure of the $\ell$-torsion of the curves (where $\ell$ is a prime different from the characteristic $p$ of the base field). Our approach is inspired by a previous algorithm due to Couveignes, which involved computations using the $p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of $\tilde{O} (r^{2})p^{O(1)}$ base field operations. On the other hand, the cost of our algorithm is $\tilde{O} (r^{2})\log (q)^{O(1)}$, for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.

MSC classification

Primary: 11Y40: Algebraic number theory computations
Secondary: 11G20: Curves over finite and local fields 14H52: Elliptic curves
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Special Issue A: Algorithmic Number Theory Symposium XII , 2016 , pp. 267 - 282
Copyright
© The Author(s) 2016 

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