Paper 2006/331

On the Security of Generalized Jacobian Cryptosystems

Isabelle Dechene


Generalized Jacobians are natural candidates to use in discrete logarithm (DL) based cryptography since they include the multiplicative group of finite fields, algebraic tori, elliptic curves as well as all Jacobians of curves. This thus led to the study of the simplest nontrivial generalized Jacobians of an elliptic curve, for which an efficient group law algorithm was recently obtained. With these explicit equations at hand, it is now possible to concretely study the corresponding discrete logarithm problem (DLP); this is what we undertake in this paper. In short, our results highlight the close links between the DLP in these generalized Jacobians and the ones in the underlying elliptic curve and finite field.

Note: Improved version of the original submission, including updated references as well as a simpler proof of Proposition 3.

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Published elsewhere. Unknown where it was published
Discrete logarithm problemgeneralized Jacobianssemi-abelian varietieselliptic curvesfinite fieldspairing-friendly curves
Contact author(s)
idechene @ uwaterloo ca
2007-03-24: revised
2006-10-05: received
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      author = {Isabelle Dechene},
      title = {On the Security of Generalized Jacobian Cryptosystems},
      howpublished = {Cryptology ePrint Archive, Paper 2006/331},
      year = {2006},
      note = {\url{}},
      url = {}
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