椭圆曲线群的标量乘快速算法研究

Fast algorithm of scalar multiplication in elliptic curve group

摘要:(摘要内容经过系统自动伪原创处理以避免复制,下载原文正常,内容请直接查看目录。)

随同着信息技巧的高速成长,信息平安成绩日趋凸起。暗码技巧是完成信息、隐蔽、完全性验证、身份认证的有用门路,是确保信息体系平安的症结技巧之一。椭圆曲线暗码(ECC)因为其平安性高、密钥短等长处,特殊合适运用于存储空间、带宽、功耗等受限情况中。椭圆曲线暗码自提出以来惹起暗码学界普遍存眷与运用,成为今朝最有前程的一种公钥暗码体系体例,获得人们普遍存眷。盘算速度是椭圆曲线暗码研讨与运用中最关怀的一个成绩,若何高效完成椭圆曲线暗码是信息平安范畴最近几年来研讨的一个热门。椭圆曲线上标量乘运算效力决议了椭圆曲线暗码运算速度,是进步椭圆曲线暗码运算速度的症结。本文深刻剖析研讨了椭圆曲线的标量乘算法,并在此基本上对标量乘的有关算法停止了改良。重要的研讨内容和研讨成果以下:(1)经由过程研讨剖析标量乘Shamir-NAF算法,对现有算法中存在的缺乏有了根本懂得:该计划中将标量k表现成NAF情势,由NAF性质可知,表现成NAF情势的标量k能够比二进制表现长度年夜1;而且在k的NAF情势中,假如能使0的情势加倍集中,可以用滑动技巧进步运算效力。针对这些缺乏,本文提出了改良的Shamir-NAF算法,改良后的算法可以下降k的表现长度,数据注解改良后的算法可以进步年夜约22%的效力。解释改良后的算法要优于原算法。(2)剖析了直接盘算办法的算法,针对该算法中标量k在位数较年夜时,好比k=128位,运算量将是伟大的缺点,提出了一种二进制表现的直接盘算办法算法。因为经由过程运用二进制对原算法停止处置,处理了k位数年夜时不实用的缺点。而且和传统NAF算法停止了剖析比较,数据注解新的二进制表现的直接盘算办法算法可以进步年夜约8%的倍点运算效力。

Abstract:

Along with the rapid growth of information technology, information security issues become increasingly prominent. The password techniques is complete information, hidden and complete the verification and authentication useful opportunities, is to ensure that one of the crux techniques of information system security. Elliptic curve cipher (ECC) because of its safe high, key short wait for an advantage, special suitable for use in the limited storage space, bandwidth and power consumption. Elliptic curve password, since it is cause code academic widespread concern and application, become the most promising a public key cipher system style, people widespread concern. Computing speed is elliptic curve cipher research and application in the care of a performance, how efficient completion of the elliptic curve password is information safety domain in recent years to research a hot. Elliptic curve scalar multiplication operation effect resolution of elliptic curve cipher computing speed, is the crux of the progress of elliptic curve cipher computing speed. This paper deeply analyses and studies the elliptic curve scalar multiplication algorithm, the algorithm and this is basically to stop the improvement of scalar multiplication. Important research content and research achievements are as follows: (1) through research analysis of scalar multiplication algorithm Shamir-NAF, lack of existing algorithms in the presence of a who knows the ultimate: the program will be the scalar K into NaF situation, NaF properties show that expression of NaF situation of the scalar K can enough than the binary length of the eve of the 1; and in K of the NaF situation, if can make the situation of 0 double concentrated, can slide technique progress operation effect. According to the deficiency, this paper proposes the Shamir-NAF algorithm improved, the improved algorithm can decrease the expression of K in length, the effectiveness of the improved algorithm of data annotation can be improved about 22%. Explain the improved algorithm is superior to the original algorithm. (2) analysis of the direct method of calculating algorithm, the algorithm for scalar K in the number of large, like the k=128, the computation will be great shortcomings, puts forward the method of calculating the direct algorithm of a binary expression. Because through the process of using binary processing to the original algorithm, processing K digit Nianye not practical disadvantages. And the traditional NAF algorithm is analyzed, the method of calculating the direct effect of point multiplication algorithm of binary data annotation new progress can be approximately 8%.

目录:

摘要6-7
Abstract7
第1章 绪论10-15
    1.1 研究背景及意义10-12
    1.2 椭圆曲线标量乘的研究现状12-13
    1.3 本论文的研究内容及章节安排13-15
第2章 椭圆曲线密码体制理论基础15-26
    2.1 群与有限域15-18
        2.1.1 群15
        2.1.2 有限域15-18
    2.2 椭圆曲线数学基础18-25
        2.2.1 椭圆曲线概念18-22
        2.2.2 GF(p)域上算术运算22-24
        2.2.3 GF(2~n)域上算术运算24-25
    2.3 本章小结25-26
第3章 椭圆曲线多标量乘改进算法26-44
    3.1 投影坐标下椭圆曲线群运算26-31
        3.1.1 雅可比坐标下椭圆曲线群运算26-30
        3.1.2 标准投影坐标下椭圆曲线群运算30-31
    3.2 标量乘算法31-36
        3.2.1 二进制算法31-32
        3.2.2 非相邻表示法(NAF)32-34
        3.2.3 直接算法34-35
        3.2.4 Shamir算法35
        3.2.5 Shamir-NAF算法35-36
    3.3 多标量乘算法的改进36-42
        3.3.1 改进Shamir-NAF算法36-41
        3.3.2 算法性能分析41-42
    3.4 本章小结42-44
第4章 基于直接计算方法的改进算法44-54
    4.1 直接计算形式方法44-46
    4.2 二进制直接计算表示法46-50
        4.2.1 DR算法分析46-49
        4.2.2 改进的DR算法49-50
    4.3 改进后算法的性能分析50-53
        4.3.1 正确性分析50
        4.3.2 算法性能分析50-53
    4.4 本章小结53-54
总结与展望54-55
致谢55-56
参考文献56-60
攻读硕士学位期间发表的论文及科研成果60