椭圆系统

ECC system makes use of the finite group of elliptic curve on the finite field instead of finite cyclic group used in the discrete logarithm problem. The theory of ECC is being studied a lot and it is the focus of the cryptography and the industrial estate, especially the technique of the implementation of ECC.In this thesis, the meaning of the information technology is discussed firstly, then we analyze the classical cryptography algoritham. After that, the mathematics base of elliptic curve cryptosytems are introduced, especially the theory of the finite field. As for some algorithms, we also give out the program of reliazation.

随后，介绍了研究椭圆曲线密码系统所需要的数学理论基础，特别是有限域中的椭圆曲线理论，并对相关的方法给出相应的实例及其程序实现；论文第三章讨论给出了椭圆曲线密码体制的研究现状，综述了椭圆曲线与现有一些密码算法的结合，介绍了椭圆曲线的离散对数问题以及椭圆曲线面临的攻击，详细描述了椭圆曲线密码系统的实现细节及各个部分的实验数据。

Second, we consider the strongly coupled elliptic system with homogeneous Dirichlet boundary conditions. The prior estimates for thesolution of nonlinear elliptic system are derived. It is shown that there is no coexistence state if diffusion rates are strong, or if the intrinsic growth rates are slow.

第二章研究带齐次Dirichlet边界条件的强耦合椭圆系统，首先推出非线性椭圆系统解的先验估计，然后证明了当食饵和捕食者的扩散率足够大，或者出生率足够小时，系统不存在共存现象，并给出半平凡解存在的充分条件。

This article first introduces the math foundation required by ECC,including the addition rule for elliptic curve point defined over finite field.Then , the principle of ECC is discussed and its security and efficiency of ECC are analyzed.Third, a cryptosystem is designed through analyzing the security requiration, choosing the elliptic curve domain parameters,denoting field element,elliptic curve and elliptic curve point,choosing associate primitves and schemes andpartitioning functional module.Forth, how to develop a crytosystem based on elliptic curve encryption algorithm is investigated.Fifth, a cryptosystem we have developed by us and the testing result is described.

本文首先介绍了ECC的数学基础，对有限域上椭圆曲线点的运算规则进行了详细描述；其次探讨了ECC的原理，分析了ECC的安全性和有效性；第三，设计了一个基于ECC的加密系统，包括系统的安全需求分析，域参数选择，域元、椭圆曲线、点的表示，原语和方案的选择，及整个系统的模块功能划分；第四，在设计的基础上，研究如何开发一个基于椭圆曲线的加密系统；第五，描述了一个我们已经设计与开发的基于椭圆曲线的加密系统，并给出了相应的测试结果。

This dissertation investigates the construction of pseudo-random sequences (pseudo-random numbers) from elliptic curves and mainly analyzes their cryptographic properties by using exponential sums over rational points along elliptic curves. The main results are as follows:(1) The uniform distribution of the elliptic curve linear congruential generator is discussed and the lower bound of its nonlinear complexity is given.(2) Two large families of binary sequences are constructed from elliptic curves. The well distribution measure and the correlation measure of order k of the resulting sequences are studied. The results indicate that they are "good" binary sequences which give a positive answer to a conjecture proposed by Goubin et al.(3) A kind of binary sequences from an elliptic curve and its twisted curves over a prime field F_p. The length of the sequences is 4p. The "1" and "0" occur almost the same times. The linear complexity is at least one-fourth the period.(4) The exponential sums over rational points along elliptic curves over ring Z_ are estimated and are used to estimate the well distribution measure and the correlation measure of order k of a family of binary sequences from elliptic curves over ring Z_.(5) The correlation of the elliptic curve power number generator is given. It is proved that the sequences produced by the elliptic curve quadratic generator are asymptotically uniformly distributed.(6) The uniform distribution of the elliptic curve subset sum generator is considered.(7) We apply the linear feedback shift register over elliptic curves to produce sequences with long periods. The distribution and the linear complexity of the resulting sequences are also considered.

本文研究利用椭圆曲线构造的伪随机序列，主要利用有限域上椭圆曲线有理点群的指数和估计讨论椭圆曲线序列的密码性质——分布、相关性、线性复杂度等，得到如下主要结果：(1)系统讨论椭圆曲线-线性同余序列的一致分布性质，即该类序列是渐近一致分布的，并给出了它的非线性复杂度下界；(2)讨论两类由椭圆曲线构造的二元序列的"良性"分布与高阶相关性(correlation of order κ)，这两类序列具有"优"的密码性质，也正面回答了Goubin等提出的公开问题；(3)利用椭圆曲线及其挠曲线构造一类二元序列，其周期为4p(其中椭圆曲线定义在有限域F_p上)，0-1分布基本平衡，线性复杂度至少为周期的四分之一；(4)讨论了剩余类环Z_上的椭圆曲线的有理点的分布估计，并用于分析一类由剩余类环Z_上椭圆曲线构造的二元序列的伪随机性；(5)讨论椭圆曲线-幂生成器序列的相关性及椭圆曲线-二次生成器序列的一致分布；(6)讨论椭圆曲线-子集和序列的一致分布；(7)讨论椭圆曲线上的线性反馈移位寄存器序列的分布，线性复杂度等性质。

In order to refine a method for defence against at least one attack made by means of differential power analysis on at least one hyperelliptic cryptosystem, in particular at least one hyperelliptic public key cryptosystem, which is given by at least one hyperelliptic curve of any genus over a finite field in a first group, where the hyperelliptic curve is given by at least one co-efficient, so that an essential contribution can be made towards an efficient and secure implementation of the hyperelliptic cryptosystem, it is proposed that the hyperelliptic curve and/or at least one element of the first group, in particular at least one in particular reduced divisor and/or at least one intermediate result of a scalar multiplication, is randomised.

为了改进用于防御在至少一个超椭圆密码系统中、特别是在至少一个超椭圆公共密钥密码系统中通过微分功率分析作出的至少一个攻击的方法，该超椭圆公共密钥密码系统是通过在第一组中有限域中任何种类的至少一个超椭圆曲线给出的，其中超椭圆曲线由至少一个系数给出，以使得可以对于超椭圆密码系统的有效而安全的实施方案作出重要的贡献，提出将超椭圆曲线和/或第一组的至少一个元素、特别是至少一个特定的减小的除数和/或标量乘法的至少一个中间结果进行随机化。

Under some conditions we obtain the existence of nontrivial solution by analyzing the best Sobolev embedding constant and by use of the minimax theorems without conditions.

通过使用没有条件的极小极大定理，以及对最佳Sobolev嵌入常数的详细分析，得到了一些具临界Sobolev指数的半线性椭圆系统的真正非平凡解的存在性，并讨论了解的一些性质。

The macroprogram is written by using the elliptical parametric equation, then some actual work problem can be solved with the macroprogram.

在数控编程的实际应用中，常见的是直线插补和圆弧插补，没有椭圆插补，但在实际生产中，往往遇到许多椭圆的零件需要加工，如果手工常规编程就无法编制出椭圆的加工程序，为了解决这个问题，在数控编程中有宏程序可以做到椭圆的加工，本文按HCNC-21T系统（在其他系统只不过是宏程序的格式不同而已）对椭圆做内、外的加工编程举例，还在内椭圆的加工中先加工一个工艺孔，巧妙利用椭圆的参数方程来编写宏程序，用宏程序来解决一些实际的工作问题。

This paper first analyses and summarizes the ststus quo and evolution trend of encryption, some common used cryptograph are introduced, including the algorithms used in symmetric cryptosystem and asymmetirc cryptosystem. We describe the theory of each algorithms and compare the elliptic curve cryptosystem with the other two asymmetric cryptosystems to show the advantages of this algorithm. Second, the principle of ECC is discussed, including the math foundation of ECC, basic conception of elliptic curves, constructiong idea of ECC, operation on the elliptic curve and so on. Third, the current attacks of ECC were analyzed deeply, and an algorithm based on limited prime number field was constructed. We analyzed its realizability in theory, and implement it by using certain function of MIRACL software package. Latter half in this paper, the implementation model of a simple elliptic curve encryption system which based on GF has been introduced. The paper also put a deep analysis on the algorithm of point addition and point multiplication.

本文首先对密码技术的发展现状及其发展趋势进行了分析和综述，详细的介绍了私钥密码系统和公钥密码系统的发展，说明各种算法的原理和优缺点，并给出了一些典型的密码体制的简要分析，重点将椭圆曲线算法与其它几种公钥密码算法比较，说明椭圆曲线算法的优势；其次，探讨了椭圆曲线密码体制的原理，包括椭圆曲线密码的数学基础、基本概念、椭圆曲线密码体制的构造思想等问题；第三作者对椭圆曲线的攻击现状作了详细的分析，针对所使用的大素数域F_p，设计了素数域上安全椭圆曲线产生的算法，从理论上做了可实施性分析，从软件上做了具体实现；在本文的后半部分，提出了一个简单的基于有限素数域上的椭圆曲线加密方按算实现模型，并对SECES中设计的点加和点乘运算进行了深入分析。

In Chapter Ⅲ, the global stability and global attractivity of some planar autonomous systems are dealed with.

第二章考虑平面自治系统同宿轨族的一些定性性态，证明了系统至多存在一个最大椭圆扇形，得到此系统的轨线趋于奇点及同宿轨族、闭轨族、双曲扇形和椭圆扇形的存在性与不存在性的充分或充要条件。

This dissertation investigates the construction of pseudo-random sequences (pseudo-random numbers) from elliptic curves and mainly analyzes their cryptographic properties by using exponential sums over rational points along elliptic curves. The main results are as follows:(1) The uniform distribution of the elliptic curve linear congruential generator is discussed and the lower bound of its nonlinear complexity is given.(2) Two large families of binary sequences are constructed from elliptic curves. The well distribution measure and the correlation measure of order k of the resulting sequences are studied. The results indicate that they are "good" binary sequences which give a positive answer to a conjecture proposed by Goubin et al.(3) A kind of binary sequences from an elliptic curve and its twisted curves over a prime field F_p. The length of the sequences is 4p. The "1" and "0" occur almost the same times. The linear complexity is at least one-fourth the period.(4) The exponential sums over rational points along elliptic curves over ring Z_ are estimated and are used to estimate the well distribution measure and the correlation measure of order k of a family of binary sequences from elliptic curves over ring Z_.(5) The correlation of the elliptic curve power number generator is given. It is proved that the sequences produced by the elliptic curve quadratic generator are asymptotically uniformly distributed.(6) The uniform distribution of the elliptic curve subset sum generator is considered.(7) We apply the linear feedback shift register over elliptic curves to produce sequences with long periods. The distribution and the linear complexity of the resulting sequences are also considered.

本文研究利用椭圆曲线构造的伪随机序列，主要利用有限域上椭圆曲线有理点群的指数和估计讨论椭圆曲线序列的密码性质——分布、相关性、线性复杂度等，得到如下主要结果：(1)系统讨论椭圆曲线-线性同余序列的一致分布性质，即该类序列是渐近一致分布的，并给出了它的非线性复杂度下界；(2)讨论两类由椭圆曲线构造的二元序列的&良性&分布与高阶相关性(correlation of order κ)，这两类序列具有&优&的密码性质，也正面回答了Goubin等提出的公开问题；(3)利用椭圆曲线及其挠曲线构造一类二元序列，其周期为4p(其中椭圆曲线定义在有限域F_p上)，0-1分布基本平衡，线性复杂度至少为周期的四分之一；(4)讨论了剩余类环Z_上的椭圆曲线的有理点的分布估计，并用于分析一类由剩余类环Z_上椭圆曲线构造的二元序列的伪随机性；(5)讨论椭圆曲线-幂生成器序列的相关性及椭圆曲线-二次生成器序列的一致分布；(6)讨论椭圆曲线-子集和序列的一致分布；(7)讨论椭圆曲线上的线性反馈移位寄存器序列的分布，线性复杂度等性质。

Plunder melds and run with this jewel!

掠夺melds和运行与此宝石！

My dream is to be a crazy growing tree and extend at the edge between the city and the forest.

此刻，也许正是在通往天国的路上，我体验着这白色的晕旋。