Jacobi Quartic曲线上GLV/GLS标量乘算法

翁江; 姬伟峰; 吴玄; 李映岐; 张林锋; 孟浩

doi:10.12263/DZXB.20191005

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Jacobi Quartic曲线上GLV/GLS标量乘算法

学术论文 | 更新时间：2025-12-08

- Jacobi Quartic曲线上GLV/GLS标量乘算法
- GLV/GLS Scalar Multiplication on Jacobi Quartic Curves
- 电子学报 2021年49卷第9期页码：1783-1789
- 作者机构：
  
  1.西安电子科技大学网络与信息安全学院, 陕西西安 710071
  2.空军工程大学信息与导航学院, 陕西西安 710077
  3.北京市海淀区复兴路14号院10分队, 北京 100089
- 作者简介：
  
  [ "翁　江　男，1986年3月出生，陕西西安人. 现为空军工程大学信息与导航学院讲师，主要研究方向为网络密码和椭圆曲线密码. E-mail: wengjiang858@163.com" ]
- 基金信息：
  
  国家自然科学基金(61902426);中国博士后科学基金第69批面上项目(2021M692502)
- DOI：10.12263/DZXB.20191005
  中图分类号： TP309
- 收稿：2019-09-03，
  
  修回：2021-01-24，
  
  纸质出版：2021-09-25
- 稿件说明：
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翁江,姬伟峰,吴玄等.Jacobi Quartic曲线上GLV/GLS标量乘算法[J].电子学报,2021,49(09):1783-1789.

WENG Jiang,JI Wei-feng,WU Xuan,et al.GLV/GLS Scalar Multiplication on Jacobi Quartic Curves[J].ACTA ELECTRONICA SINICA,2021,49(09):1783-1789.
翁江,姬伟峰,吴玄等.Jacobi Quartic曲线上GLV/GLS标量乘算法[J].电子学报,2021,49(09):1783-1789. DOI： 10.12263/DZXB.20191005.

WENG Jiang,JI Wei-feng,WU Xuan,et al.GLV/GLS Scalar Multiplication on Jacobi Quartic Curves[J].ACTA ELECTRONICA SINICA,2021,49(09):1783-1789. DOI： 10.12263/DZXB.20191005.

摘要

目前GLV/GLS (Gallant，Lambert，Vanstone / Galbraith

Lin

Scott)标量乘算法的研究主要集中在Weierstrass曲线上，尝试寻找和构造更多或者更高次数的可有效计算的自同态.本文主要研究了Jacobi Quartic曲线上GLV/GLS标量乘算法.首先利用曲线之间的双有理等价，给出了该类曲线在素域上可有效计算自同态的具体构造，得到2维GLV方法.然后考虑椭圆曲线的二次扭曲线，利用曲线之间双有理等价和Frobenius映射，给出了该类曲线在二次扩域上可有效计算自同态的具体构造，得到2维GLS方法.将上述GLV和GLS方法结合起来，同时利用曲线在二次扩域上的两个不同的自同态，得到4维GLV方法.最后针对

不变量为0或1728两类特殊形式的椭圆曲线，利用更高次的扭曲线，得到4维GLV方法.实验结果表明：对于Jacobi Quartic曲线，2维GLV方法和4维GLV方法比5-NAF方法分别提速37.2%和109.4%以上.同时，在三种不同的实现方式下，Jacobi Quartic曲线上标量乘效率都优于Weierstrass曲线.

Abstract

At present

GLV/GLS scalar multiplication mainly focuses on the Weierstrass curves

attempting to find and construct more and more efficient computable endomorphism. In this paper

we study the applications of GLV/GLS method on Jacobi Quartic curve. Firstly

we present the concrete construction of efficiently computable endomorphism for this type of curves over prime field by exploiting birational equivalence between curves

and obtain 2-dimensional GLV method. Secondly

we consider the quadratic twists of elliptic curves. By using birational equivalence and Frobenius mapping between curves

we present methods to construct efficiently computable endomorphisms of this type of curves over the quadratic extension field

and obtain a 2-dimensional GLS method. Finally

we obtain the 4-dimensional GLV method on elliptic curves with

-invariant 0 or 1728 by using higher degree twists. The experimental results show that the speedups of 2-dimensional GLV method and 4-dimensional GLV method than 5-NAF method exceed 37.2% and 109.4% for Jacobi Quartic curves respectively. At the same time

under the three implementations above

the scalar multiplication on the Jacobi Quartic curves is always more efficient than that on the Weierstrass curves.

关键词

Keywords

references

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