学校编码 :10384 分类号密级 学号 : UDC 硕士学位论文 椭圆曲线公钥密码体系实现技术的研究 The Research of Implementation Technology Based on Elliptic Curve Cryptosystem and Public
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1 学校编码 :10384 分类号密级 学号 : UDC 硕士学位论文 椭圆曲线公钥密码体系实现技术的研究 The Research of Ipleentation Technology Based on Elliptic Curve Cryptosyste and Public Key Cryptography 贾英涛 指导教师姓名 : 郑建德教授 专业名称 : 计算机应用技术 论文提交日期 :2007 年 4 月 论文答辩时间 :2007 年月 学位授予日期 :2007 年月 答辩委员会主席 : 评阅人 : 2007 年 5 月
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3 厦门大学学位论文原创性声明 兹呈交的学位论文, 是本人在导师指导下独立完成的研究成果 本人在论文写作中参考的其他个人或集体的研究成果, 均在 文中以明确方式标明 本人依法享有和承担由此论文产生的权利 和责任 声明人 ( 签名 ): 年月日
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5 厦门大学学位论文著作权使用声明 本人完全了解厦门大学有关保留 使用学位论文的规定 厦门大学有权保留并向国家主管部门或其指定机构送交论文的纸质版和电 子版, 有权将学位论文用于非赢利目的的少量复制并允许论文进入学 校图书馆被查阅, 有权将学位论文的内容编入有关数据库进行检索, 有权将学位论文的标题和摘要汇编出版 保密的学位论文在解密后适 用本规定 本学位论文属于 1 保密( ), 在 年解密后适用本授权书 2 不保密( ) ( 请在以上相应括号内打 ) 作者签名 : 日期 : 年月日 导师签名 : 日期 : 年月日
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7 摘要 摘要 椭圆曲线密码系统 (ECC) 建立在椭圆曲线群上离散对数 (ECDLP) 的难解性这一数学难题 与其他公钥密码系统相比, 椭圆曲线密码系统除了安全性高外, 还 具有计算负载小, 密钥尺寸短, 占用带宽少等优点 IEEE ANSI ISO IETF SECG 等组织已在椭圆曲线密码算法的标准化方面做了大量工作, 分别形成了自 己的标准, 椭圆曲线密码系统大有替代 RSA 成为主流公钥密码体制的趋势 因此, 研究椭圆曲线公钥密码系统的实现技术具有很大的现实意义 本文第二章首先介绍关于椭圆曲线密码系统中用到的群 环 域以及相关的中国剩余定理等有关数学基础知识, 然后详细阐述整个椭圆曲线密码系统, 重点介绍基于素数域和特征为 2 的有限域上的椭圆曲线上点加 倍加 点乘等运算 最后介绍安全椭圆曲线的攻击算法, 阐述了攻击算法中用到的攻击技术理论, 并指出安全椭圆曲线参数选取时的注意事项和几种选取算法 论文第三章一方面提出椭圆曲线密码算法中基于二进制表示的点乘算法的半滑动窗口改进算法, 通过进行基于素数域 GF ( p) 上的椭圆曲线密码算法中点乘算法的实验, 分析比较改进算法的效率 另一方面详细介绍基于椭圆曲线的 ECDSA 数字签名算法, 并基于 iracl 库对其进行了基于素数域 GF ( p) 的实现 论文第四章分析了单一认证的弱安全性, 结合 PKCS#11 标准和 J2EE 平台的 EJB 组件技术, 设计并实现了基于用户口令 (PIN) 认证和硬件 USBKey 数字证书身份认证的双重因素认证系统 最后讨论了椭圆曲线密码系统比基于 RSA 的数字 签名系统的优越性, 并分析了将椭圆曲线密码系统应用于双因素认证系统的可行 性 论文第五章对本论文进行了总结和对以后工作的展望 关键词 : 椭圆曲线 ; 点乘算法 ; 双因素认证 ;
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9 Abstract Abstract Elliptic Curve Cryptosyste (ECC) is based on the Matheatical probles which is called elliptic curve discrete logarith proble (ECDLP). Copared with other public-key cryptographies, ECC not only has the higher security but also has less coputation overheads, shorter key size and narrower bandwidth. Soe organizations such as IEEE, ANSI, ISO, IETF SECG, have done a lot of the standardization work on algorith of Elliptic Curve Cryptosyste and each has developed its own standard. The experts believe that ECC will becoe the next generation popular public key cryptography. Therefore, the study of the Elliptic Curve Cryptosyste technology has great practical significance. In the chapter 2, we first introduce the background and basic atheatic concepts related to Elliptic Curve Cryptosyste such as group, ring and field.and the whole Elliptic Curve Cryptosyste is analyzed in detail. Then the basic operation such as point add is presented based on GF( p) or GF (2 ). Next, the algorith of attacks to the ECDLP is analyzed, and the security criterion is proposed. The security criterion can prevent the elliptic curve fro being attacked. At the sae tie, soe algorith to generate secure elliptic curve is listed. In the chapter 3, the scalar ultiplication algorith first is introduced. Then the iproved algorith of scalar ultiplication called half sliding window algorith is proposed. It is based on binary representation of ultiplier. Next, we collect the result datu throw several experients and represent the efficiency with chart using atlab. At last, the Digital Standard Algorith of ECC is ipleented. It is based on GF( p) and used the library of iracl. In chapter 4, the insecurity of single PIN is analyzed. Then a dual
10 Abstract factor authentication syste is designed. It cobines Public Key Crypto Standard #11(PKCS#11) with EJB coponent technology on the basis of PIN and UsbKey digital certificate. Finally, copared with RSA,the Advantage of ECC is discussed.then We analyze the feasibility that applying ECC on the dual factor authentication syste. In the last chapter, the conclusion and deeper work are presented. Key Words: Elliptic Curve Cryptosyste (ECC); Scalar Multiplication Algorith; Dual Factor Authentication.
11 目录 目录 第一章引言 密码系统概述 私钥密码系统概述 公钥密码系统概述 椭圆曲线密码体系 椭圆曲线密码体系的优势 椭圆曲线密码体系的标准 本文研究目标与内容 研究目标 研究内容... 8 第二章椭圆曲线密码体系的数学理论与概述 群 (Groups) 环 (Ring) 域(Field) 素数域及其运算 特征为 2 的有限域 GF (2 ) 及其运算 其它数论基础 模运算 素数与 Miller-Rabin 素性测试 最大公因子和欧几里德算法 中国剩余定理 平方剩余 勒让德符号 雅可比符号 椭圆曲线简介 实数域上的椭圆曲线 素数域 GF ( p) 上的椭圆曲线 特征为 2 的有限域 GF (2 ) 上的椭圆曲线 GF (2 ) 上椭圆曲线分类 GF (2 ) 上椭圆曲线加法规则 对现有椭圆曲线的攻击算法 对一般曲线的攻击 对特殊曲线的攻击 安全椭圆曲线的选取与算法分类...23 第三章椭圆曲线上公钥密码体系及其实现技术... 25
12 目录 3.1 数字签名标准 数字签名概述 数字签名的形式化定义与分类 DSA 数字签名算法 椭圆曲线上公钥密码系统的实现技术 点的坐标表示 点加和倍加 点乘 椭圆曲线上 DSA 数字签名系统实现技术 椭圆曲线公钥密码系统要素 基于 ECDSA 的密钥的生成 ECDSA 签名的生成 ECDSA 签名的验证 系统实现软硬件环境 系统使用说明 本章小结 第四章 ECDSA 在双因素认证系统中的应用 双因素认证系统分析 双因素认证系统设计 安全认证协议设计 UML 建模认证过程序列图 J2EE 平台 RSA 算法的双因素认证系统实现 PKCS#11 标准 通用模型 UsbKey 硬件签名构架 UsbKey 硬件签名实现 EJB 组件的验证核心部分实现 系统测试数据结果输出 椭圆曲线上签名算法在双因素认证系统的上应用前景 椭圆曲线密码体制的优越性 椭圆曲线密码体制应用于双因素认证系统的可行性分析 本章结论 第五章总结与展望...59 参考文献...61 攻读硕士学位期间发表学术论文情况...65 致谢...67
13 Contents Contents Chapter 1 Introduction Cryptosyste Introduction Private Key Cryptosyste Introduction Public key Cryptosyste Introduction Elliptic Curve Cryptosyste Advantage of ECC Standard of ECC Research Targets and Contents Research Targets Research Contents...8 Chapter 2 Matheatic theory and Introduction of EC Groups Ring and Field Operating on GF( p ) Operating on GF (2 ) Other Theoriy Modular Arithetic Prie Nuber and Miller-Rabin Test Greatest Coon Divisor and Euclid Algorith Chinese Reainder Theore Square Reainder Legendre Sybol Jacobi Sybol Elliptic Curve Elliptic Curve on Real Nuber Field Elliptic Curve on GF ( p) Elliptic Curve on GF (2 ) Elliptic Curve Classes on GF (2 ) Addition on Elliptic Curve based on GF (2 ) Attack Algoriths Attacking Coon Elliptic Curve Attacking Single Elliptic Curve Choosing Secure Elliptic Curve and Algorith Classes Chapter 3 Introduction and Ipleentation of ECC... 25
14 Contents 3.1 Digital Signature Standard Introduction of Digital Signature Foralized Defination and Types of Digital Signature Digital Signature Algorith Ipleentation Techniques of PKI on ECC Point Representation on Different Coordinates Point Addtion and Doubling Point Multiplation Ipleentation Techniques of DSA on ECC Eleents of PKI on ECC Generating the Private Key of ECDSA Generating Signature Generating Signature Software and Hardware of Syste Specification of Syste Suary Chapter 4 Applaction of RSA and ECDSA on Dual Factor Authentication Syste Dual Factor Authentication Syste Analysis Dual Factor Authentication Syste Design Secure Authentication Protocol Design Sequence Diagra of Authentication Process on UML Ipleent of Dual Factor Authentication Syste Based J2EE Platfor PKCS#11 Standard Coon Model Model of UsbKey Signature Ipleent of UsbKey Signing EJB Coponent of Verifying Testing Data Foreground of Dual Factor Authentication Syste Using ECC Advantage of ECC Feasibility Analysis Suary Chapter 5 Suary and Further Research The Referrence Literatures Publications Acknowledgent...67
15 第一章引言 第一章引言 随着计算机科学 通信和网络技术的迅速向前发展, 信息化己成为社会发展 的趋势, 人们也开始将一些现实行为在网络上实现, 例如 : 网上银行, 网上投票 等 然而由于网络本身的开放性和脆弱性, 使得网络信息面临日益严重的威胁和 攻击, 信息安全问题也日益严重和突出 伴随着社会的各个部门对网络安全性要 求的越来越高, 特别是一些涉及机密性的机构和金融机构对更高安全性的需求, 使得信息安全的作用越来越明显 因此, 设计好的密码系统也变得至关重要 1.1 密码系统概述一个密码系统的主角一般有三方, 即发送方, 接收方与攻击方 ( 密码分析者 ), 典型的密码系统如图 1-1 所示 在发送方, 首先将明文 M 利用加密器 E 及加密密钥 Ke, 将明文加密成密文 C = E(M,Ke) 接着将 C 利用公开信道送给接收方, 接收方收到密文 C 后, 利用解密器 D 及解密密钥 Kd, 将 C 解密成明文 M = D(C,Kd) 明文 发送方 加密器 (Encryptor) 加密密钥 公开信道 秘文 C=E(M,Ke) 攻击方 接收方 解密器 (Decryptor) 解密密钥 Ke Kd 图 1-1 典型密码系统方框图 一般而言, 密码系统依其应用可对信息提供下列功能 : (1) 秘密性 : 防止非法的接收者发现明文 (2) 鉴别性 : 确定信息来源的合法性, 也即此信息确实是由发送方所传送, 1
16 椭圆曲线公钥密码体系实现技术的研究 而非别人伪造 (3) 完整性 : 确定信息没有被有意或无意的更改, 及被部分取代 加入或删除等等 (4) 不可否认性 : 发送方在事后不可否认其传送过的信息 现代密码学系统按密钥形式分为私钥密码系统和公钥密码系统 私钥密码系 统是收发双方加 / 解密的过程所使用的密钥都相同的密码系统, 又叫作对称密码 系统, 传统的密码系统都属此类 公钥密码系统是指收发双方加 / 解密的过程所 使用的密钥不相同的密码系统, 又叫作非对称密码系统 1.2 私钥密码系统概述 1949 年 Shannon 发表的 保密系统的信息理论 为私钥密码系统建立了理论基础, 从此密码学成为一门科学 私钥密码系统的优点在于密码算法简便, 加密速度快, 保密度高 ; 它的最大问题是密钥的分发和管理非常复杂 代价高昂 比如对于具有 n 个用户的网络, 需要 nn ( 1)/2个密钥, 在用户群不是很大的情况下, 对称加密系统是有效的 但是对于大型网络, 当用户群很大, 分布很广时, 密钥的分配和保存就成了大问题 另外, 由于通信双方必须统一密钥, 才能发送保密的信息 如果发信者与收信人是素不相识的, 就无法向对方发送秘密信息了 常用的私钥密码算法有以下几种 : (1)DES 算法 DES(Data Encryption Standard) 是由 IBM 公司研制的, 美国国家标准局公布的一种分组对称密码算法 自公布以来, 它一直超越国界成为国际上商 用保密通信和计算机通信的最常用的加密算法 DES 算法使用长度为 56 比特的 密钥加密长度为 64 比特的明文, 获得长度为 64 比特的密文 鉴于其密钥长度比较短, 随着硬件的发展, 已经不再安全 (2) 三重 DES 算法三重 DES 算法用两个密钥对明文进行三次加密, 假设两个密钥是 K1 和 K2, 加密步骤如下 : <1> 用密钥 K1 进行 DES 加密 <2> 用 K2 对步骤 <1> 的结果进行 DES 解密 <3> 用步骤 <2> 的结果使用密钥 K1 进行 DES 加密 2
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