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Journal of Mathematical Cryptology

Managing Editor: Magliveras, Spyros S. / Steinwandt, Rainer / Trung, Tran

Editorial Board: Blackburn, Simon R. / Blundo, Carlo / Burmester, Mike / Cramer, Ronald / Gilman, Robert / Gonzalez Vasco, Maria Isabel / Grosek, Otokar / Helleseth, Tor / Kim, Kwangjo / Koblitz, Neal / Kurosawa, Kaoru / Lauter, Kristin / Lange, Tanja / Menezes, Alfred / Nguyen, Phong Q. / Pieprzyk, Josef / Rötteler, Martin / Safavi-Naini, Rei / Shparlinski, Igor E. / Stinson, Doug / Takagi, Tsuyoshi / Williams, Hugh C. / Yung, Moti

CiteScore 2017: 1.43

SCImago Journal Rank (SJR) 2017: 0.293
Source Normalized Impact per Paper (SNIP) 2017: 1.117

Mathematical Citation Quotient (MCQ) 2017: 0.51

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Volume 5, Issue 1


An exploration of affine group laws for elliptic curves

Huseyin Hisil / Kenneth Koon-Ho Wong / Gary Carter / Ed Dawson
Published Online: 2011-04-10 | DOI: https://doi.org/10.1515/jmc.2011.005


Several forms of elliptic curves are suggested for an efficient implementation of Elliptic Curve Cryptography. However, a complete description of the group law has not appeared in the literature for most popular forms. This paper presents group law in affine coordinates for three forms of elliptic curves. With the existence of the proposed affine group laws, stating the projective group law for each form becomes trivial. This work also describes an automated framework for studying elliptic curve group law, which is applied internally when preparing this work.

Keywords.: Elliptic curve; group law; point addition; point doubling; projective coordinates; rational maps; birational equivalence; Riemann–Roch theorem; rational simplification; scalar multiplication; elliptic curve cryptography

About the article

Received: 2010-09-04

Revised: 2011-03-03

Published Online: 2011-04-10

Published in Print: 2011-06-01

Citation Information: Journal of Mathematical Cryptology, Volume 5, Issue 1, Pages 1–50, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: https://doi.org/10.1515/jmc.2011.005.

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