International Association
for Cryptologic Research

This paper proposes the computation of the Tate pairing,

Ate pairing and its variations on the special Jacobi quartic elliptic curve

Y^2 = dX^4 +Z^4. 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 [2] and

previous results on Jacobi quartic curves [3]. 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

ecient, comparatively to the case of Weierstrass curves with quartic

twists [2].