Tate Pairing Computation on Jacobi’s Elliptic Curves

  • Sylvain Duquesne
  • Emmanuel Fouotsa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7708)


We propose for the first time the computation of the Tate pairing on Jacobi intersection curves. For this, we use the geometric interpretation of the group law and the quadratic twist of Jacobi intersection curves to obtain a doubling step formula which is efficient but not competitive compared to the case of Weierstrass curves, Edwards curves and Jacobi quartic curves. As a second contribution, we improve the doubling and addition steps in Miller’s algorithm to compute the Tate pairing on the special Jacobi quartic elliptic curve Y 2 = dX 4 + Z 4. 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 all the curves with quartic twists. In particular for the doubling step in Miller’s algorithm, we obtain a theoretical gain between 6% and 21%, depending on the embedding degree and the extension field arithmetic, with respect to Weierstrass curves [6] and Jacobi quartic curves [23].


Jacobi quartic curves Jacobi intersection curves Tate pairing Miller function group law geometric interpretation birational equivalence 


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sylvain Duquesne
    • 1
  • Emmanuel Fouotsa
    • 2
  1. 1.IRMAR, UMR CNRS 6625Université Rennes 1Rennes cedexFrance
  2. 2.Département de MathématiquesUniversité de Yaoundé 1, Faculté des SciencesYaoundéCameroun

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