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November  2007, 1(4): 413-426. doi: 10.3934/amc.2007.1.413

On the security of generalized Jacobian cryptosystems

Isabelle Déchène 1,

1. 

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada

Received  March 2007 Revised  August 2007 Published  October 2007

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.
Keywords: semi-abelian varieties, generalized Jacobians, Public-key cryptography, finite fields, discrete logarithm problem, elliptic curves, pairing-friendly curves..
Mathematics Subject Classification: Primary: 14L35, 94A60; Secondary: 68Q1.
Citation: Isabelle Déchène. On the security of generalized Jacobian cryptosystems. Advances in Mathematics of Communications, 2007, 1 (4) : 413-426. doi: 10.3934/amc.2007.1.413
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