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Identifying supersingular elliptic curves

Part of: Arithmetic algebraic geometry Computational number theory Curves

Published online by Cambridge University Press:  01 September 2012

Andrew V. Sutherland
Andrew V. Sutherland*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, MA 02139, Cambridge, USA (email: drew@math.mit.edu)

Abstract

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Given an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits structural differences between ordinary and supersingular isogeny graphs. This yields a simple algorithm that, given E and a suitable non-residue in 𝔽p2, determines the supersingularity of E in O(n3log 2n) time and O(n) space, where n=O(log p) . Both these complexity bounds are significant improvements over existing methods, as we demonstrate with some practical computations.

MSC classification

Secondary: 11G07: Elliptic curves over local fields 11Y16: Algorithms; complexity 11G20: Curves over finite and local fields 14H52: Elliptic curves
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 317 - 325
Copyright
Copyright © London Mathematical Society 2012

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