Abstract:

So far, pairing computation is implemented on elliptic curves in the plane, such as the Weierstrass, Edwards and Jacobi quartic curves. This paper discusses pairing computation on elliptic curves in the threedimensional space for the first time. As elliptic curves in the threedimensional space for cryptography, intersections of quadric surfaces have important relations with the Edwards curves and the Jacobi quartic curves, which gives a deep comprehension of the Edwards curves and the Jacobi quartic curves. For simplicity, we just consider pairing computation on the Jacobi intersections. However, our results can be generalized to other intersections of the quadric surfaces. We first analyze the geometric properties of the Jacobi intersections and construct efficiently computable endomorphisms for the Jacobi intersections. Finally, we give pairing computation and optimization for the Jacobi intersections.

Key words: elliptic curves;pairing computation;Jacobi intersections;Miller algorithm;efficiently computable endomorphisms