Property and Scalar Multiplication Algorithm on Supersingular Elliptic Curves with j Invariant 0

WENG Jiang; KANG Xiao-chun; DOU Yun-qi; MA Chuan-gui

doi:10.3969/j.issn.0372-2112.2018.09.013

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Property and Scalar Multiplication Algorithm on Supersingular Elliptic Curves with j Invariant 0

更新时间：2025-07-16

- Property and Scalar Multiplication Algorithm on Supersingular Elliptic Curves with j Invariant 0
- Acta Electronica Sinica Vol. 46, Issue 9, Pages: 2131-2138(2018)
- 作者机构：
  
  1. 空军工程大学信息与导航学院,陕西,西安,710077
  2. 中国传媒大学信息工程学院,北京,100024
  3. 数学工程与先进计算国家重点实验室,河南,郑州,450001
  4. 陆军航空兵学院基础部,北京,101123
  5. 空军工程大学信息与导航学院,陕西,西安,710077
  6. 中国传媒大学信息工程学院,北京,100024
  7. 数学工程与先进计算国家重点实验室,河南,郑州,450001
  8. 陆军航空兵学院基础部,北京,101123
- 作者简介：
- 基金信息：
  
  National Natural Science Foundation of China (No.61379150);Subject supported by Open Fund of State Key Laboratory of Mathematical Engineering and Advanced Computing (No.2016A02);Key Technology Research and Development Program of Henan Province (No.122102210126, No.092101210502)
- DOI：10.3969/j.issn.0372-2112.2018.09.013
  CLC： TP309
- Published Online：25 September 2018，
  
  Published：2018
- 稿件说明：
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WENG Jiang, KANG Xiao-chun, DOU Yun-qi, et al. Property and Scalar Multiplication Algorithm on Supersingular Elliptic Curves with j Invariant 0[J]. Acta Electronica Sinica, 2018, 46(9): 2131-2138.
DOI：

WENG Jiang, KANG Xiao-chun, DOU Yun-qi, et al. Property and Scalar Multiplication Algorithm on Supersingular Elliptic Curves with j Invariant 0[J]. Acta Electronica Sinica, 2018, 46(9): 2131-2138. DOI： 10.3969/j.issn.0372-2112.2018.09.013.

摘要

针对非超奇异椭圆曲线上的标量乘算法已经有比较多的研究.与非超奇异曲线不同，超奇异椭圆曲线的自同态环是四元数代数的一个序模，为非交换环.本文主要针对特征大于3的有限域上一类

不变量为0的超奇异椭圆曲线，分析了曲线自同态环及其商环的结构.进而研究了此类曲线上整数表示的性质，并基于这种表示方法提出了一种针对此类曲线的标量乘算法.理论上证明了针对此类超奇异曲线，当选择合适系数集合时，此表示实质上为

-adic展开.实验结果表明：相较于4-NAF等方法，

-adic表示方法提高标量乘效率一倍以上.

Abstract

The scalar multiplication algorithms for non-supersingular elliptic curves have been widely studied. In contrast

the endomorphism ring of supersingular elliptic curve is an order in a definite quaternion algebra

which is not commutative. This paper focuses on a class of supersingular elliptic curves of

-invariant zero in characteristic greater than 3. We make analysis of the structures of its endomorphism ring and quotient ring. Further we study the properties of integer expansion according to this class of curves. Based on this representation

a scalar multiplication algorithm is proposed. We demonstrate that the representation is essentially the

-adic expansion in theory when a suitable d

igit set is chosen. The experimental results show that compared with the method of 4-NAF

the

-adic method improves the efficiency of scalar multiplication of more than 100%.

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