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,...

Generalized Jacobians are natural candidates to use in discrete logarithm (DL) based cryptography since they include the multiplicative group of …nite …elds, algebraic tori, elliptic curves as well as all Jacobians of curves. This observation therefore raised the possibility that other families of generalized Jacobians may be of cryptographic interest. This exploration began with the study of t...

Déchène has proposed generalized Jacobians as a source of groups for public-key cryptosystems based on the hardness of the Discrete Logarithm Problem (DLP). Her specific proposal gives rise to a group isomorphic to the semidirect product of an elliptic curve and a multiplicative group of a finite field. We explain why her proposal has no advantages over simply taking the direct product of group...

In this paper, three key theorems (the open mapping theorem, the inverse function theorem, and the implicit function theorem) for continuously differentiable maps are shown to hold for nonsmooth continuous maps which are not necessarily Lipschitz continuous. The significance of these extensions is that they are given using generalized Jacobians, called approximate Jacobians. The approximate Jac...