On the security of generalized Jacobian cryptosystems

Isabelle Déchène

doi:10.3934/amc.2007.1.413

Abstract

Generalized Jacobians are natural candidates to use in discretelogarithm (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 anelliptic 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.

Highlights

Throughout this past year, cryptographers proudly celebrated three decades of public-key cryptography
In addition of being the ...rst such problem to be used in public-key cryptography, the discrete logarithm problem (DLP) remains without a doubt one of the most popular choices used nowadays to design cryptographic protocols
We demonstrate that the DLP in such cyclic generalized Jacobians is at least as hard as the DLP in E and at least as hard as the DLP in F

Summary

Introduction

Throughout this past year, cryptographers proudly celebrated three decades of public-key cryptography. Public-key cryptography, discrete logarithm problem, generalized Jacobians, semi-abelian varieties, elliptic curves, ...nite ...elds, pairing-friendly curves. Galbraith and Smith [6] recently made similar observations by working with extensions of algebraic groups “presented by a cocycle” This more general setting extends some of the results of this paper, we believe that our explicit approach has the advantage of being more insightful. These techniques enabled us to highlight an apparent distinct behaviour of generalized Jacobians of pairing-friendly curves (see Section 6 for more details).

Generalized Jacobians of an Elliptic Curve

A Natural Solution

Reductions among Discrete Logarithm Problems

A Parallel Solution à la Pohlig-Hellman

Generalized Jacobians of Pairing-friendly Curves

Outlook