Faster Pairing Computation on Jacobi Quartic Curves with High-Degree Twists

  • Fan Zhang
  • Liangze Li
  • Hongfeng WuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9473)


In this paper, we first propose a geometric approach to explain the group law on Jacobi quartic curves which are seen as the intersection of two quadratic surfaces in space. Using the geometry interpretation we construct Miller function. Then we present explicit formulae for the addition and doubling steps in Miller’s algorithm to compute the Tate pairing on Jacobi quartic curves. Our formulae on Jacobi quartic curves are better than previously proposed ones for the general case of even embedding degree. Finally, we present efficient formulas for Jacobi quartic curves with twists of degree 4 or 6. Our pairing computation on Jacobi quartic curves are faster than the pairing computation on Weierstrass curves when \(j=1728\). The addition steps of our formulae are fewer than the addition steps on Weierstrass curves when \(j=0\).


Elliptic curve Jacobi quartic curve Tate pairing Miller function Group law 



This work was supported by National Natural Science Foundation of China (No. 11101002, No. 11271129 and No. 61370187), Beijing Natural Science Foundation (No. 1132009), and the General Program of Science and Technology Development Project of Beijing Municipal Education Commission of China.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.LMAM, School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.Beijing International Center for Mathematical ResearchBeijingChina
  3. 3.College of SciencesNorth China University of TechnologyBeijingChina

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