Summary: | Groups where the discrete logarithm problem (DLP) is believed to be intractable have proved to be inestimable building blocks for cryptographic applications. They are at the heart of numerous protocols such as key agreements, public-key cryptosystems, digital signatures, identification schemes, publicly verifiable secret sharings, hash functions and bit commitments. The search for new groups with intractable DLP is therefore of great importance. The study of such a candidate, the so-called generalized Jacobians, is the object of this dissertation. The motivation for this work came from the observation that several practical discrete logarithm-based cryptosystems, such as ElGamal, the Elliptic and Hyperelliptic Curve Cryptosystems, XTR, the Lucas-based cryptosystem LUC as well as the torus-based cryptosystem CEILIDH can all naturally be reinterpreted in terms of generalized Jacobians. We next provide, from a cryptographic point of view, a global description of this family of algebraic groups that highlights their potential for applications. Our main contribution is then to introduce a new public-key cryptosystem based on the simplest non-trivial generalized Jacobian of an elliptic curve. This work thus provides the first concrete example of a semi-abelian variety suitable for DL-based cryptography.
|