paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve . 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 and , 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 more efficient compared to the case of Weierstrass curves with
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.