Jacobi四次曲线的快速差分加法公式

Efficient Differential Addition Formulae on Jacobi Quartic Curves

椭圆曲线的点乘运算是各类椭圆曲线密码体系中的关键运算,Montgomery算法是计算椭圆曲线点乘的有效算法之一,它能够有效地抵抗简单能量分析.Jacobi四次曲线具有良好的密码学属性,和其它椭圆曲线模型相比,Jacobi四次曲线上的点乘运算具有很好的效率优势.定义在有限域上的每个偶数阶椭圆曲线都双有理等价于一个Jacobi四次曲线.本文提出了Jacobi四次曲线上的快速差分加法公式.在射影坐标系统下,本文提出的混合加法和倍乘运算的总花费仅需要5M+4S+1D或者3M+6S+3D,其中M、S和D分别表示有限域上的乘法运算,平方运算和常数乘法运算.相较于Jacobi四次曲线上的已有结果,本文提出的公式是目前最有效的.本文的结果进一步提升了Jacobi四次曲线模型的竞争力.

The point multiplication is a key operation in elliptic curve cryptographic schemes.Montgomery algorithm is an efficient point multiplication algorithm on elliptic curve.It has effective resistance against simple power analysis.Jacobi quartic curve has good cryptographic properties.Compared with other elliptic curve models,the point multiplication operation on Jacobi quartic curve has good efficiency advantages.Every even order elliptic curve defined over a finite field is birational equivalent to a Jacobi quartic curve.This paper presents explicit differential addition and doubling formulae for Jacobi quartic curve over finite fields.In this paper,a fast differential addition formula on a general Jacobi quartic curve is designed.Using the projective coordinate system,the total cost of differential addition and doubling in this paper only needs 5M+4S+1D and 3M+6S+3D operations respectively,where M,S,and D denote the fixed costs of a field multiplication,a field squaring and a field multiplication,respectively.The cost of differential addition is lower than that of all previously proposed methods.The results of this paper further enhance the competitiveness of Jacobi quartic curve model.