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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 / Dawson, Ed / 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

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CiteScore 2017: 1.43

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

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Volume 5, Issue 3-4


Families of elliptic curves with rational 3-torsion

Dustin Moody
  • National Institute of Standards and Technology – Computer Security Division, 100 Bureau Drive Stop 8930, Gaithersburg, Maryland 20899-8930, USA
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/ Hongfeng Wu
Published Online: 2012-01-25 | DOI: https://doi.org/10.1515/jmc-2011-0013


In this paper we look at three families of elliptic curves with rational 3-torsion over a finite field. These families include Hessian curves, twisted Hessian curves, and a new family we call generalized DIK curves. We find the number of -isogeny classes of each family, as well as the number of -isomorphism classes of the generalized DIK curves. We also include some formulas for efficient computation on these curves, improving upon known results. In particular, we find better formulas for doubling and addition on the original tripling-oriented DIK curves and also for addition and tripling on elliptic curves with -invariant 0.

Keywords.: Elliptic curves; Hessian curves; cryptography

About the article

Received: 2011-09-02

Revised: 2011-12-07

Accepted: 2011-12-08

Published Online: 2012-01-25

Published in Print: 2012-02-01

Citation Information: Journal of Mathematical Cryptology, Volume 5, Issue 3-4, Pages 225–246, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: https://doi.org/10.1515/jmc-2011-0013.

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