Paper 2006/033

Arithmetic of Generalized Jacobians

Isabelle Déchène


This paper aims at introducing generalized Jacobians as a new candidate for discrete logarithm (DL) based cryptography. The motivation for this work came from the observation that several practical DL-based cryptosystems, such as ElGamal, the Elliptic and Hyperelliptic Curve Cryptosystems, XTR, LUC as well as CEILIDH can all naturally be reinterpreted in terms of generalized Jacobians. However, usual Jacobians and algebraic tori are thus far the only generalized Jacobians implicitly utilized in cryptography. In order to go one step further, we here study the simplest nontrivial generalized Jacobians of an elliptic curve. In this first of a series of articles, we obtain explicit formulae allowing to efficiently perform arithmetic operations in these groups. This work is part of our doctoral dissertation, where security aspects are considered in depth. As a result, these groups thus provide the first concrete example of semi-abelian varieties suitable for DL-based cryptography.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Unknown where it was published
Discrete logarithm problemgeneralized Jacobianssemi-abelian varietieselliptic curves
Contact author(s)
idechene @ uwaterloo ca
2006-01-31: received
Short URL
Creative Commons Attribution


      author = {Isabelle Déchène},
      title = {Arithmetic of Generalized Jacobians},
      howpublished = {Cryptology ePrint Archive, Paper 2006/033},
      year = {2006},
      note = {\url{}},
      url = {}
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