Tate Pairing Computation on Jacobi’s Elliptic Curves

Abstract

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].