Efficient computation of pairings on Jacobi quartic elliptic curves

  • 1 IRMAR, UMR CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
  • 2 Laboratoire d'Informatique Avancé de Saint-Denis, Université Paris 8, 93526 Saint Denis cedex, France
  • 3 Department of Mathematics, Higher Teacher Training College, University of Bamenda, P.O. Box 5052, Bamenda, Cameroon

Abstract

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.

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JMC is a forum for original research articles in the area of mathematical cryptology. Works in the theory of cryptology and articles linking mathematics with cryptology are welcome. Submissions from all areas of mathematics significant for cryptology are published, including but not limited to, algebra, algebraic geometry, coding theory, combinatorics, number theory, probability and stochastic processes.

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