Simpler and Faster Pairings from the Montgomery Ladder

Giacomo Pope; Krijn Reijnders; Damien Robert; Alessandro Sferlazza; Benjamin Smith

Paper 2025/672

Simpler and Faster Pairings from the Montgomery Ladder

Giacomo Pope

, NCC Group, University of Bristol

Krijn Reijnders

, KU Leuven

Damien Robert

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

Alessandro Sferlazza

, Technical University of Munich

Benjamin Smith

, Inria Saclay - Île-de-France Research Centre, Computer Science Laboratory of the École Polytechnique, French National Centre for Scientific Research, Institut Polytechnique de Paris, École Polytechnique

Abstract

We show that Montgomery ladders compute pairings as a by-product, and explain how a small adjustment to the ladder results in simple and efficient algorithms for the Weil and Tate pairing on elliptic curves using cubical arithmetic. We demonstrate the efficiency of the resulting cubical pairings in several applications from isogeny-based cryptography. Cubical pairings are simpler and more performant than pairings computed using Miller's algorithm: we get a speed-up of over 40% for use-cases in SQIsign, and a speed-up of about 7% for use-cases in CSIDH. While these results arise from a deep connection to biextensions and cubical arithmetic, in this article we keep things as concrete (and digestible) as possible. We provide a concise and complete introduction to cubical arithmetic as an appendix.

Metadata

Available format(s): PDF
Category: Implementation
Publication info: Preprint.
Keywords: pairings elliptic curves isogeny-based cryptography
Contact author(s): giacomopope @ gmail com
crypto krijn @ gmail com
damien robert @ inria fr
alessandro sferlazza @ tum de
smith @ lix polytechnique fr
History: 2025-04-15: approved; 2025-04-14: received; See all versions
Short URL: https://ia.cr/2025/672
License: CC BY

BibTeX

@misc{cryptoeprint:2025/672,
      author = {Giacomo Pope and Krijn Reijnders and Damien Robert and Alessandro Sferlazza and Benjamin Smith},
      title = {Simpler and Faster Pairings from the Montgomery Ladder},
      howpublished = {Cryptology {ePrint} Archive, Paper 2025/672},
      year = {2025},
      url = {https://eprint.iacr.org/2025/672}
}