The geometric interpretation of the Tate pairing and its applications

Damien Robert

Paper 2023/177

The geometric interpretation of the Tate pairing and its applications

Damien Robert

, Inria Bordeaux - Sud-Ouest Research Centre, Institut de Mathématiques de Bordeaux

Abstract

While the Weil pairing is geometric, the Tate pairing is arithmetic: its value depends on the base field considered. Nevertheless, the étale topology allows to interpret the Galois action in a geometric manner. In this paper, we discuss this point of view for the Tate pairing: its natural geometric interpretation is that it gives étale $μ_{n}$ -torsors. While well known to experts, this interpretation is perhaps less known in the cryptographic community. As an application, we explain how to use the Tate pairing to study the fibers of an isogeny, and we prove a conjecture by Castryck and Decru on multiradical isogenies.

Note: Gave more examples of 'pairing the volcano' situation

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Preprint.
Keywords: pairings isogenies
Contact author(s): damien robert @ inria fr
History: 2023-05-13: last of 4 revisions; 2023-02-12: received; See all versions
Short URL: https://ia.cr/2023/177
License: CC BY

BibTeX

@misc{cryptoeprint:2023/177,
      author = {Damien Robert},
      title = {The geometric interpretation of the Tate pairing and its applications},
      howpublished = {Cryptology {ePrint} Archive, Paper 2023/177},
      year = {2023},
      url = {https://eprint.iacr.org/2023/177}
}