椭圆曲线密码研究与实现

【作者】 罗涛；

【导师】 易波；

【作者基本信息】 湖南大学 ， 计算机应用技术， 2003， 硕士

【摘要】 椭圆曲线密码体制是目前公钥体制中每比特密钥安全强度最高的一种密码体制，在相同安全强度条件下，椭圆曲线密码体制具有较短的密钥长度，较少的计算量、存储量、带宽等优点，而且椭圆曲线密码体制已经被许多国际标准化机构作为标准化文件向全球颁布，被认为是下一代最通用的公钥密码系统。 本文首先介绍并分析了椭圆曲线密码体制的优点及研究现状；其次研究了椭圆曲线密码体制的基本理论；第三，分析了椭圆曲线密码的安全性并介绍了密钥共享，加密，数字签名等椭圆曲线密码体制；第四，深入研究了特征为2的有限域F₂m中的元素在多项式基和最优正规基表示下的乘法运算和乘法逆运算的快速算法，并对Hankerson等人提出的多项式基下的乘法运算的快速算法作了改进，而且在实验的基础上不仅分析研究了F₂m域中元素在多项式基和最优正规基表示下的乘法和乘法逆运算的性能，还对这两种基表示下的F₂m域中元素运算效率的优劣作了比较和研究，所得的结论可供在实现椭圆曲线密码体制时参考；第五，研究了目前流行的计算椭圆曲线标量乘法的快速算法，同时改进了固定基点梳形法，提高了整个系统的速度，并在实验的基础上分析研究了流行算法的优劣；第六，实现了基于F₂m的椭圆曲线密码体制的算法库，在我们的算法库中只需稍微改变便能实现基于任意尺寸的F₂m上的ECDH，ECES，ECDSA等椭圆曲线密码体制；第七，实现了两条安全椭圆曲线上的椭圆曲线密码体制，包括ECDH，ECES，ECDSA。一条是ANSIX9.62中的191bit的安全曲线，F₂m中的元素用多项式基表示，一条是我们安全曲线库中的148bit的安全曲线，F₂m中的元素用最优正规基表示；最后对本文进行总结和展望了以后的工作。更多还原

【Abstract】 The highest safety strength of private key per bit in the Public-Key Cryptography systems is the Elliptic Curve Cryptography at present. Under similar secure conditions, the ECC has the advantages such as: less computation amounts, shorter length of private key, smaller storing and bandwidth. Moreover, it has been declared as standard documents adopted by many international standard institutions and regarded as the most universally used public key system.Firstly, we generalize and analyze the advantages and present research of Elliptic Curve Cryptography; secondly, we study the basic theory of the ECC; thirdly, we illustrate the safety of the ECC and discuss the Elliptic Curve Key Agreement Scheme, Elliptic Curve Encryption Scheme and Elliptic Curve Digital Signature Algorithm; fourthly, we study fast algorithms of the multiplication and inversion multiplication of the element of in the underlying finite field F2m whose characteristic is two represented by the two basis of optimal normal basis and polynomial basis. We make improvements to the fast algorithm of the polynomial basis multiplication by Hankerson and base on the experiments, we describe the properties and compare the advantages of the multiplication and inversion multiplication of the elements in F2m field under optimal normal bases and polynomial basis .Results concluding from the study car be used as references in the realization of the elliptic curve cryptosystem; fifthly, we overview the current fast algorithm of point multiplication, improve the fix base point comb algorithm, advance the speed of the whole system and remark the advantages and disadvantages of the popular algorithms based upon the experimental datas; sixthly we realize the algorithm library of elliptic curve cryptography based on the F2m. Only change slightly in our algorithm library can we realize the ECDH, ECES, ECDSA based onF2m of anysize; seventhly, we realize the ECC on two secure elliptic curves, including ECDH, ECES, ECDSA. One is the 191 bits secure curve of ANSI X9.62 , elements in F2m field represented by polynomial basis, the other is the 148bits secure curve in our secure curve library elements in F2m field represented by optimal normal basis; the conclusion summarizes the whole paper and preview the the further developments of the work we have done.更多还原