Group Law of Elliptic Curves over Z<sub>n</sub> and Its Application in Paillier Schemes

ZHANG Li-jun; WANG Kun-peng

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Group Law of Elliptic Curves over Z_n and Its Application in Paillier Schemes

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- Group Law of Elliptic Curves over Z_n and Its Application in Paillier Schemes
- Acta Electronica Sinica Vol. 39, Issue 8, Pages: 1733-1738(2011)
- 作者机构：
  
  中国科学院研究生院信息安全国家重点实验室,北京,100049
- 作者简介：
- 基金信息：
- DOI：
  CLC： TN918.1
- Published：2011
- 稿件说明：
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ZHANG Li-jun, WANG Kun-peng. Group Law of Elliptic Curves over Z_n and Its Application in Paillier Schemes[J]. Acta Electronica Sinica, 2011, 39(8): 1733-1738.
DOI：

ZHANG Li-jun, WANG Kun-peng. Group Law of Elliptic Curves over Z_n and Its Application in Paillier Schemes[J]. Acta Electronica Sinica, 2011, 39(8): 1733-1738. DOI：

摘要

本文研究了定义在环

上椭圆曲线

的有理点集

(

)的群的定义法则

以及这类椭圆曲线在密码学上的应用.我们给出了环

上椭圆曲线明确的点加公式

使得

E(Z

)形成一个交换群

从理论上清晰地解决了以往在该类曲线上某些点相加不能定义的问题.为了在密码学上方便地应用这类椭圆曲线

还给出了在该加法法则下的

(

)的群结构.利用该群结构选取合适的参数

修复了椭圆曲线Paillier密码系统的安全缺陷.

Abstract

Group law of rational points

(

) on elliptic curves

over

and application of this group law are studied in this paper.We present explicit addition formulae to make

(

) an abelian group.This solves the problem that some points addition can not be defined using other addition laws in the literature.For the convenient cryptographic application of such elliptic curves

we give the group structure of

(

) under our addition law.We propose a remedy for the original elliptic curves Paillier schemes which have security defectiveness.

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Group Law of Elliptic Curves over Zn and Its Application in Paillier Schemes

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摘要

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Group Law of Elliptic Curves over Z_n and Its Application in Paillier Schemes