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.%目前已知的配对计算都是在椭圆曲线的平面模型下实现的,比如Weierstrass型曲线、Ed-wards曲线和Jacobi四次曲线.本文第一次讨论空间曲线上配对的具体计算.密码学中所关心的空间曲线主要是三维空间中的二次曲面的交,它与Edwards曲线、Jacobi四次型都有极其紧密的联系,因而研究二次曲面交上的算术与配对将促进我们对Edwards曲线、Jacobi四次曲线上的相关特性的理解.为了讨论的简洁,我们将主要分析Jacobi交,但我们的结果基本上可以类推到其他的二次曲面交上去.我们分析了Jacobi交上的几何特性,构造了Jacobi交上的有效可计算同态,并在此基础上给出了Jacobi交上配对的具体计算.
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