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摘要人们对信息传递可靠性日益增长的需求，促使公钥密码算法的研究成果也不断完善和推陈出新，作为当前公钥密码体制中重要的一个领域，椭圆曲线公钥算法（ECC）具有良好的可靠性和广阔的发展前景．
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基于此，本文主要用来研究 ECC的理论基础和算法结构．全文共分为四个部分：第一部分作为绪论，对密码学，尤其是现代密码学的发展历程和理论模型作了一些总结，并对当前ECC的基础――椭圆曲线上离散对数问题（ECDLP）研究的现状，及SM2 算法的概况做了简要介绍．第二部分介绍了群、有限域及其相关理论、运算设定的一些简单知识，这些知识足以用来理解有限域上椭圆曲线的构造和ECC 的加、解密的运行原理．第三部分首先介绍了椭圆曲线的来源及其上抽象加法运算，继而将椭圆曲线置于有限域背景下，由此引出 ECDLP 的具体定义，给出不易被攻克的椭圆曲线的选择标准，最后叙述了ECC的运作流程，以及 SM2公钥算法的实现步骤．第四部分根据SM2 算法的思想，实现了一个以字节为单位的加、解密程序，对其算法结构和设计细节作了解释和总结．22291
毕业论文关键词：公钥密码学 椭圆曲线 加密 MFC
Research and implementation of the elliptic curve public-key cryptography for SM2 algorithm
Abstract
The reliability of the transmission for information is growing gradually. It makes
the research of public-key cryptography algorithm continue to improve and bring
forth the new through the old. As an important field for public-key cryptosystem
at present, the Elliptic Curve Cryptography (ECC) has good reliability and broad
prospects for development.
This paper mainly used to research the theory and the algorithm structure of
ECC.The full text is made up of four parts: The first one is introduction,
including cryptography, especially the development history and theoretical
model of modern cryptography, then give a brief introduce for the theoretical
basis of ECC -- elliptic curve discrete logarithm problem (ECDLP) , and the
situation of SM2 algorithm are also introduced in here. The second part
introduces some simple knowledge of the group, finite field and some related
theory and operation set to them. With these knowledge we are enough to
understand the tectonic theory of elliptic curves over finite fields and the
encryption-decryption operation principle of ECC. The third part introduces the
origin of abstraction addition on elliptic curve firstly, and put the elliptic curve into
finite field background, then define the ECDLP; after that some selection
standards are listed, those make the elliptic curve are not easy to be crack;
finally giving the algorithms operation process for ECC and SM2. The fourth
part implement a encryption program which is according to the thought of SM2
algorithm, and detailed interpretation for the algorithm structure and design details are summarized.
Keywords public-key cryptography elliptic curve encryption MFC
目 次
1 绪论 ． 1
1.1 密码学概述 1
1.2 椭圆曲线公钥算法的相关研究进展 4
1.3 SM2公钥算法的设定与评价 5
2 有限域与相关理论概述 ． 7
2.1 群及相关定义 7
2.2 域、 有限域及相关定义 ． 7
2.3 有限域上的运算 8
2.4 素性测试 ． 10
3 椭圆曲线加密算法概述 11