Pairing computation and arithmetic of elliptic curves for cryptography
type: Dataset
description: While first used to solve the Discrete Logarithm Problem (DLP) in the group of points of elliptic curves, bilinear pairings are now useful to construct many public key protocols. The efficiency of pairings computation depends on the arithmetic of the model chosen for the elliptic curve and of the base field where the curve is defined. In this thesis, we compute and implement pairings on elliptic curves of Jacobi forms and we study the arithmetic of a new Edwards model for elliptic curves defined over any finite field. More precisely, We use the geometric interpretation of the group law of Jacobi intersection curves to obtain the first explicit formulas for the Miller function in Tate pairing computation in this case. For pairing computation with even embedding degree, we define and use the quadratic twist of this curve to obtain efficient formulas in the doubling and addition stages in Miller's algorithm. Moreover, for pairing computation with embedding degree divisible by 4 on the special Jacobi quartic elliptic curve Ed :Y²=dX⁴+Z⁴, we define and use its quartic twist to obtain a best result with respect to Weierstrass curves. Our result is at the same time an improvement of a result recently obtained on this curve, and is therefore, to our knowledge, the best result to date on Tate pairing computation among all curves with quartic twists. In 2006, Hess et al. introduced the concept of Ate pairing which is an improving version of the Tate pairing. We extend the computation of this pairing and its variations to the curve E_d. Again our theoretical results show that this curve offers the best performances comparatively to other curves with quartic twists, especially Weiertrass curves. As a third contribution, we introduce a new Edwards model for elliptic curves with equation 1+x²+y²+x²y²=\lambda xy. This model is ordinary over binary fields and we show that it is birationally equivalent to the well known Edwards model x²+y²=c²(1+x²y²) over non-binary fields. For this…
landing page: https://theses.hal.science/tel-00919779
identifier: NNT: 2013REN1S070
keyword: couplages
keyword: courbes-elliptiques
keyword: cryptographie
keyword: cryptography
keyword: edwards-model
keyword: elliptic-curves
keyword: fonctions-theta
keyword: infoeu-reposemanticsdoctoralthesis
keyword: jacobi-model
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: modele-de-jacobi
keyword: modelededwards
keyword: pairings
keyword: theses
keyword: theta-functions
issued: 2026-05-07T18:46:33.170513
modified: 2026-05-07T18:46:33.170521
language: en
contact point: Organization
  fn: CCSD

test_moissonnage_selune
type: Agent

Pairing computation and arithmetic of elliptic curves for cryptography
type: Distribution
format: HTML
access URL: https://theses.hal.science/tel-00919779
issued: 2026-05-07T18:46:33.179818
modified: 2026-05-07T18:46:33.156384