Binary edwards curves

Daniel J. Bernstein, Tanja Lange, Reza Rezaeian Farashahi

Research output: Chapter in Book/Report/Conference proceedingConference proceeding contribution

76 Citations (Scopus)

Abstract

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 . 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, Proceedings
EditorsElisabeth Oswald, Pankaj Rohatgi
Pages244-265
Number of pages22
Volume5154 LNCS
DOIs
Publication statusPublished - 2008
Event10th International Workshop on Cryptographic Hardware and Embedded Systems, CHES 2008 - Washington, D.C., United States
Duration: 10 Aug 200813 Aug 2008

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5154 LNCS
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other10th International Workshop on Cryptographic Hardware and Embedded Systems, CHES 2008
CountryUnited States
CityWashington, D.C.
Period10/08/0813/08/08

Keywords

  • Binary fields
  • Complete addition law
  • Countermeasures against side-channel attacks
  • Edwards curves
  • Elliptic curves
  • Montgomery ladder

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  • Cite this

    Bernstein, D. J., Lange, T., & Rezaeian Farashahi, R. (2008). Binary edwards curves. In E. Oswald, & P. Rohatgi (Eds.), Cryptographic Hardware and Embedded Systems - CHES 2008 - 10th International Workshop, Proceedings (Vol. 5154 LNCS, pp. 244-265). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5154 LNCS). https://doi.org/10.1007/978-3-540-85053-3_16