Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields

Curves and Jacobians over function fields

Arithmetic of Generalized Jacobians

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,...

Legendre elliptic curves over finite fields

Throughout this paper, q > 1 denotes a power of an odd prime number p, and k is a field. Given two elliptic curves E/k and E′/k, all morphisms from E to E′ are understood to be defined over k. In particular, we simply write End(E) for the ring of all endomorphisms of E/k. The notation E ≃ E′ indicates that E is isomorphic to E′, and E ∼ E′ means that E and E′ are isogenous. The endomorphism of ...

Arithmetic over Function Fields

These notes accompany lectures presented at the Clay Mathematics Institute 2006 Summer School on Arithmetic Geometry. The lectures summarize some recent progress on existence of rational points of projective varieties defined over a function field over an algebraically closed field.

Finiteness properties of soluble arithmetic groups over global function fields

Let G be a Chevalley group scheme and B ≤ G a Borel subgroup scheme, both defined over Z. Let K be a global function field, S be a finite non-empty set of places over K , and OS be the corresponding S–arithmetic ring. Then, the S– arithmetic group B(OS) is of type F |S|−1 but not of type FP |S| . Moreover one can derive lower and upper bounds for the geometric invariants Σ(B(OS)). These are sha...