Discrete Logarithms in Generalized Jacobians

TL;DR

This paper critically examines the use of generalized Jacobians in cryptography, showing they offer no advantages over traditional groups and may have inferior security and efficiency.

Contribution

It clarifies the structure of generalized Jacobians and demonstrates their disadvantages compared to standard Jacobians in cryptographic applications.

Findings

Generalized Jacobians are isomorphic to a semidirect product of elliptic curves and finite field groups.

They offer no security or efficiency benefits over direct product groups.

Generalized Jacobians are less suitable for cryptographic use than standard Jacobians.

Abstract

D\'ech\`ene 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 groups. We then argue that generalized Jacobians offer poorer security and efficiency than standard Jacobians.