Binary Edwards curves

D.J. Bernstein, T. Lange, R. Rezaeian Farashahi

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

91 Citations (Scopus)
2 Downloads (Pure)


This paper presents a new shape for ordinary elliptic curves over fields of characteristic 2. Using the new shape, this paper presents the first complete addition formulas for binary elliptic curves, i.e., addition formulas that work for all pairs of input points, with no exceptional cases. If n = 3 then the complete curves cover all isomorphism classes of ordinary elliptic curves over F2n. This paper also presents dedicated doubling formulas for these curves using 2M+ 6S + 3D, where M is the cost of a field multiplication, S is the cost of a field squaring, and D is the cost of multiplying by a curve parameter. These doubling formulas are also the first complete doubling formulas in the literature, with no exceptions for the neutral element, points of order 2, etc. Finally, this paper presents complete formulas for differential addition, i.e., addition of points with known difference. A differential addition and doubling, the basic step in a Montgomery ladder, uses 5M+ 4S + 2D when the known difference is given in affine form.
Original languageEnglish
Title of host publicationCryptographic Hardware and Embedded Systems - CHES 2008 (10th International Workshop, Washington DC, USA, August 10-13, 2008, Proceedings)
EditorsE. Oswald, P. Rohatgi
Place of PublicationBerlin
ISBN (Print)978-3-540-85052-6
Publication statusPublished - 2008

Publication series

NameLecture Notes in Computer Science
ISSN (Print)0302-9743


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