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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Analogues of Vélu’s formulas for isogenies on alternate models of elliptic curves
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by Dustin Moody and Daniel Shumow PDF
Math. Comp. 85 (2016), 1929-1951 Request permission

Abstract:

Isogenies are the morphisms between elliptic curves and are, accordingly, a topic of interest in the subject. As such, they have been well studied, and have been used in several cryptographic applications. Vélu’s formulas show how to explicitly evaluate an isogeny, given a specification of the kernel as a list of points. However, Vélu’s formulas only work for elliptic curves specified by a Weierstrass equation. This paper presents formulas similar to Vélu’s that can be used to evaluate isogenies on Edwards curves and Huff curves, which are normal forms of elliptic curves that provide an alternative to the traditional Weierstrass form. Our formulas are not simply compositions of Vélu’s formulas with mappings to and from Weierstrass form. Our alternate derivation yields efficient formulas for isogenies with lower algebraic complexity than such compositions. In fact, these formulas have lower algebraic complexity than Vélu’s formulas on Weierstrass curves.
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Additional Information
  • Dustin Moody
  • Affiliation: Computer Security Division, National Institute of Standards and Technology (NIST), Gaithersburg, Maryland 20899
  • MR Author ID: 870964
  • Email: dustin.moody@nist.gov
  • Daniel Shumow
  • Affiliation: Microsoft Research, Redmond, Washington 98052-6399
  • MR Author ID: 1072502
  • Email: danshu@microsoft.com
  • Received by editor(s): December 16, 2013
  • Received by editor(s) in revised form: July 10, 2014, and December 23, 2014
  • Published electronically: September 9, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 85 (2016), 1929-1951
  • MSC (2010): Primary 14K02; Secondary 14H52, 11G05, 11Y16
  • DOI: https://doi.org/10.1090/mcom/3036
  • MathSciNet review: 3471114