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 Jacobians of hyperelliptic curves.

This thus led to the study of the simplest nontrivial generalized Jacobians of an elliptic curve, which is an extension (of algebraic groups) of the elliptic curve by the multiplicative group. We will first take a look at the arithmetic in these generalized Jacobians. With explicit equations at hand, we then study its discrete logarithm problem (DLP), which is at the heart of the security of DL-based cryptosystems.

No prior knowledge of cryptography will be assumed.