Paper 2018/839

On Kummer Lines With Full Rational 2-torsion and Their Usage in Cryptography

Huseyin Hisil and Joost Renes


A paper by Karati and Sarkar at Asiacrypt'17 has pointed out the potential for Kummer lines in genus one, by observing that its SIMD-friendly arithmetic is competitive with the status quo. A more recent preprint explores the connection with (twisted) Edwards curves. In this paper we extend this work and significantly simplify their treatment. We show that their Kummer line is the x-line of a Montgomery curve translated by a point of order two, and exhibit a natural isomorphism to a twisted Edwards curve. Moreover, we show that the Kummer line presented by Gaudry and Lubicz can be obtained via the action of a point of order two on the y-line of an Edwards curve. The maps connecting these curves and lines are all very simple. As an example, we present the first implementation of the qDSA signature scheme based on the squared Kummer line. Finally we present close estimates on the number of isomorphism classes of Kummer lines.

Available format(s)
Public-key cryptography
Publication info
Preprint. MINOR revision.
Montgomery curvesEdwards curvesKummer linesMontgomery ladderDigital signatures
Contact author(s)
j renes @ cs ru nl
2018-09-06: received
Short URL
Creative Commons Attribution


      author = {Huseyin Hisil and Joost Renes},
      title = {On Kummer Lines With Full Rational 2-torsion and Their Usage in Cryptography},
      howpublished = {Cryptology ePrint Archive, Paper 2018/839},
      year = {2018},
      note = {\url{}},
      url = {}
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