# Quelques aspects de l'arithmétique des courbes hyperelliptiques de genre 2

Abstract : In this report, we are interested in basic building blocs for asymmetric cryptography and mainly on the discrete logarithm problem. In the first part, we present an overview of different pairings on Jacobians of curves of genus $2$ and describe the details of a careful implementation. We make a comparison with same level of safety pairings on elliptic curves. A second part is devoted to finding effective models for elliptic curves and non-ordinary Kummer surfaces in characteristic 2. For genus one, we obtain that the binary model of Edwards is derived from the classical model of Edwards in characteristic zero. For genus $2$, we use techniques of "distortion" which consists in considering a family of Jacobians over a ring of formal series, such as the generic fiber is regular and the special fiber is considered the Jacobian. It is then to show that the group law on the generic fiber extends to the whole model. We compare the laws of composition thus obtained with those already known.
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https://tel.archives-ouvertes.fr/tel-00506025
Contributor : Oumar Diao <>
Submitted on : Monday, July 26, 2010 - 9:18:17 PM
Last modification on : Thursday, January 7, 2021 - 4:25:15 PM
Long-term archiving on: : Tuesday, October 23, 2012 - 11:25:36 AM

### Identifiers

• HAL Id : tel-00506025, version 1

### Citation

Oumar Diao. Quelques aspects de l'arithmétique des courbes hyperelliptiques de genre 2. Mathématiques [math]. Université Rennes 1, 2010. Français. ⟨tel-00506025⟩

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