基域

First, basing on point halving, this thesis proposed a new efficient algorithm of computing kP lQ on an elliptic curve over the finite field with characteristic 2 and analyzed the computational efficiency. When the ratio I/M of inversion to multiplication cost is small, this new algorithm is about 5%～30% faster than the traditional algorithms.

首先，基于半点运算给出一种计算特征为2的有限域上椭圆曲线双标量乘法kP lQ的新算法并分析了效率，当基域上求乘法与求逆的运算时间的比值比较小时，新算法的效率比已有算法提高5％～30％。

The divisor class group, often called Jacobian group, based on which hyperelliptic curve cryptosystems are constructed, is much complicated than the elliptic curve rational point group.

但由于可以在一个很小的基域上构造具有较大素数因子阶的Jacobian群，且可应用于密码体制的超椭圆曲线比较丰富，这使得超椭圆曲线密码体制也逐渐受到人们的重视。

The main contributions of the second part of this dissertation are focused on the cryptographic properties of logical functions over finite field, with the help of the properties of trace functions, and that of p-polynomials, as well as the permutation theory over finite field: The new definition of Chrestenson linear spectrum is given and the relation between the new Chrestenson linear spectrum and the Chrestenson cyclic spectrum is presented, followed by the inverse formula of logical function over finite field; The distribution for linear structures of the logical functions over finite field is discussed and the complete construction of logical functions taking on all vectors as linear structures is suggested, which leads to the conception of the extended affine functions over finite field, whose cryptographic properties is similar to that of the affine functions over field GF (2) and prime field F〓; The relationship between the degeneration of logical functions and the linear structures, the degeneration of logical functions and the support of Chrestenson spectrum, as well as the relation between the nonlinearity and the linear structures are discussed; Using the relation of the logical functions over finite field and the vector logical functions over its prime field, we reveal the relationship between the perfect nonlinear functions over finite field and the vector generalized Bent functions over its prime field; The existence or not of the perfect nonlinear functions with any variables over any finite fields is offered, and some methods are proposed to construct the perfect nonlinear functions by using the balanced p-polynomials over finite field.

重新定义了有限域上逻辑函数的Chrestenson线性谱，考察了新定义的Chrestenson线性谱和原来的Chrestenson循环谱的关系，并利用一组对偶基给出了有限域上逻辑函数的反演公式；给出了有限域上随机变量联合分布的分解式，并利用随机变量联合分布的分解式对有限域上逻辑函数的密码性质进行了研究；给出了有限域上逻辑函数与相应素域上向量逻辑函数的关系，探讨了它们之间密码性质的联系，如平衡性，相关免疫性，扩散性，线性结构以及非线性度等；讨论了有限域上逻辑函数各类线性结构之间的关系，并给出了任意点都是线性结构的逻辑函数的全部构造，由此引出了有限域上的"泛仿射函数"的概念；考察了有限域上逻辑函数的退化性与线性结构的关系、退化性与Chrestenson谱支集的关系；给出了有限域逻辑函数非线性度的定义，利用有限域上逻辑函数的非线性度与相应素域上向量逻辑函数非线性度的关系，考察了有限域上逻辑函数的非线性度与线性结构的关系；利用有限域上逻辑函数与相应素域上向量逻辑函数的关系，揭示了有限域上的广义Bent函数与相应素域上的广义Bent函数的关系，以及有限域上的完全非线性函数与相应素域上向量广义Bent函数之间的关系；给出了任意有限域上任意n元完全非线性函数存在性与否的完整证明，并利用有限域上平衡的p-多项式的性质给出了有限域上完全非线性函数的一些基本构造方法。

According to the Galois extension theory of the number field,the discriminant of the field and the integral basis theory,the necessary and sufficient condition is attained which is m≡n≡1(mod4),where m and n are two different rational integers without square factors.

四次域是重要的Galois数域之一，该文对四次域Q进行了讨论；利用Galois扩张、域的判别式及整基等一些理论，得出了其具有正规整基的充要条件为m≡n≡1(mod4)(其中m，n是2个不同的无平方因子有理整数)。

Because the hardware and software may be implemented quickly if wechoose proper normal bases to represent finite fields. In this paper, we first introducesome basic definitions and facts on normal bases over finite fields, then extend theseresults to any Galois extension fields.

在这篇文章中，我们先介绍了有限域上的一些基本定义及关于正规基的相关结论，进而将域扩张到一般的伽罗瓦扩域中，得到了伽罗瓦扩域中正规基与其对偶基等价的充分必要条件，从而将有限域上正规基的等价条件推广到任意伽罗瓦扩域上。

In particular, this proof is independent of the ground field.

特别地，此证明不依赖于基域。

At the same level of security, the basis field of hyperelliptic curve cryptosystems is smaller than that of ECC, and almost all protocols based on the standard DLP such as DSA and ElGamal can be planted to HCC.

在同等安全水平下，超椭圆曲线密码要比椭圆曲线密码所用的基域小，且HCC可以模拟基于一般乘法群上的如DSA、ElGamal等几乎所有协议；在同样的定义域上，亏格越大(g≤4)，曲线越多，所以选取用于密码中的安全曲线的余地越大。

At the same level of security, the basis field of hyperelliptic curve cryptosystems is smallerthan that of ECC, and almost all protocols based on the standard DLP such as DSA andElGamal can be planted to HCC.

在同等安全水平下，超椭圆曲线密码要比椭圆曲线密码所用的基域小，且HCC可以模拟基于一般乘法群上的如DSA、ElGamal等几乎所有协议；在同样的定义域上，亏格越大(g≤4)，曲线越多，所以选取用于密码中的安全曲线的余地越大。

Hyperelliptic curves cryptography is one kind surpassing of ellipse curve cryptography.

在同等安全水平下，HECC比ECC所用的基域小，并且在同样的定义域上，HECC可以提供更多的曲线，这样可以用于密码学中的安全的曲线的选项便增多了，由于这些优点使得HECC在近几年来，在密码界中受到越来越多的重视。

The implementation processes of those two algorithms on smart card with cryptographic coprocessor are explained separately. The implementation process of RSA includes modular exponentiation and modular multiplication. The process of ECC includes the selection of base field and coordinates, scalar multiplication and field operation.

本文分别阐述了两者在带有加密协处理器的智能卡平台上的实现过程，包括RSA算法中模幂运算、模乘运算的实现；ECC算法中基域的选择、坐标系的选择、标量乘法和域算术运算的实现。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。