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BY-NC-ND 3.0 license Open Access Published by De Gruyter July 11, 2014

Efficient computation of pairings on Jacobi quartic elliptic curves

  • Sylvain Duquesne EMAIL logo , Nadia El Mrabet and Emmanuel Fouotsa


This paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve Y2=dX4+Z4. We improve the doubling and addition steps in Miller's algorithm to compute the Tate pairing. We use the birational equivalence between Jacobi quartic curves and Weierstrass curves, together with a specific point representation to obtain the best result to date among curves with quartic twists. For the doubling and addition steps in Miller's algorithm for the computation of the Tate pairing, we obtain a theoretical gain up to 27% and 39%, depending on the embedding degree and the extension field arithmetic, with respect to Weierstrass curves and previous results on Jacobi quartic curves. Furthermore and for the first time, we compute and implement Ate, twisted Ate and optimal pairings on the Jacobi quartic curves. Our results are up to 27% more efficient compared to the case of Weierstrass curves with quartic twists.

MSC: 14H52

Funding source: ANR

Award Identifier / Grant number: 12-BS01-0010-01 “PEACE”

Funding source: INS

Award Identifier / Grant number: 2012 SIMPATIC

Funding source: LIRIMA

Award Identifier / Grant number: 2013 MACISA

The authors thank the anonymous referees and the program committee of Pairing 2012 for their useful comments on the first version of this work.

Received: 2013-9-16
Revised: 2014-4-22
Accepted: 2014-7-3
Published Online: 2014-7-11
Published in Print: 2014-12-1

© 2014 by De Gruyter

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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