分解域

In this paper, we introduce the algorithm of Schoof-Elkies-Atkin to compute the order of elliptic curves over finite fields. We give out a fast algorithm to compute the division polynomial f〓 and a primitive point of order 2〓. This paper also gives an improved algorithm in computing elliptic curve scalar multiplication. Using the method of complex multiplication, we find good elliptic curves for use in cryptosystems, and implemented ElGamal public-key scheme based on elliptic curves. As a co-product, we also realized the algorithm to determine primes using Goldwasser-Kilian's theorem. Lastly, the elliptic curve method of integer factorization is discussed. By making some improvement and through properly selected parameters, we successfully factored an integer of 55 digits, which is the product of two 28-digit primes.

本文介绍了计算有限域上椭圆曲线群的阶的Schoof-Elkies-Atkin算法，在具体处理算法过程中，我们给出了计算除多项式f〓的快速算法和寻找2〓阶本原点的快速算法；标量乘法是有关椭圆曲线算法中的最基本运算，本文对[Koe96]中的椭圆曲线标量乘法作了改进，提高了其运算速度；椭圆曲线的参数的选择直接影向到椭圆曲线密码体的安全性，文中利用复乘方法构造了具有良好密码特性的椭圆曲线，并实现了椭圆曲线上ElGamal公钥体制；文中还给出了利用Goldwasser-Kilian定理和椭圆曲线的复乘方法进行素数的确定判别算法；最后讨论了利用椭圆曲线分解整数的方法并进行了某些改进，在PC机上分解了两个28位素数之积的55位整数。

By using unified method , we investigate decomposability of real or complex cubic polynomials in several elements .

这里，本文将运用统一的方法对实数域和复数域上的多元三次多项式的分解问题加以讨论。

Bull; Decomposability: if some work has already been done to analyze the modular structure of the problem domain, it may provide a good starting point for the modular decomposition of the software.

分解性：如果某些工作已经完成了分析问题域的模块结构，那么对于软件的模块分解，它可以提供一个良好的开端。

From 1920s up to Now, various techniques and methods for wave field decomposition, propagation and migration/imaging have been well developed, such as the Kirchhoff asymptotic method, frequency-wavenumber domain phase-shift and phase-shift-plus-interpolation methods, and the one-way wave equation based phase-screen and generalized screen methods, etc.

自上个世纪二十年代至今，波场分解、传播与偏移成像技术经历了将近一个世纪的发展，形成了多种不同的方法，如Kirchhoff波动方程高频近似解方法，频率-波数域的相移(phase-shift)方法和相移-内插方法，以及在混合域中基于单向波动方程的相位屏、广义屏方法等。

It includes the basic structural theory of groups; groups'actions on sets and the applications of these actions; the basic structures of rings; the properties of uniquely factorized domains; the Chinese Remainder; the extensions of fields; finite fields with applications; the Galois theory with applications.

主要内容包括群的基本结构理论、群在集合上的作用及其应用、环的基本结构和因子分解理论、中国剩余定理、域的扩张理论、有限域及其应用、Galois 理论及其应用。

Tansformed the real part and imaginary part of interferogram to stationary wavelet domain, analyzed the PDF of signal and noise wavelet coefficients using Pearson distribution system, then evaluated the wavelet coefficients of signal in stationary wavelet domain by Maximum A Posterior criteria to reconstruct the interferogram.

2将干涉图的实部和虚部分解到静态小波域，用Pearson分布系作近似，分析了信号和噪声小波系数的概率密度函数，在静态小波域用MAP准则估计信号的小波系数用于干涉图的重建。

In three cases, we investigate the decomposition of a polynomial f=x~n -u in a local field to factor a prime ideal of K in its extension.

在这三种情况下，我们研究多项式f=x~n－u局部域上分解情况，得出相应K上素理想具体分解形式。

In the following two cases:(1)gcd=1;(2) N=p〓, where p denotes the characteristic of the finite field GF, by using the factorization of the polynomial in finite field, the counting function N〓 is showed, the expected value of the linear complexity E〓 is determined, and a useful lower bound on the expected value of the k-error linear complexity E〓 is established.(7) The stability of the lattice structure for pseudorandom number sequences is put forward.

利用多项式的因式分解等简单工具，研究了任意有限域GF上，周期N与p互素以及N ＝P〓这两种情况下，计数函数N〓，的值，即线性复杂皮为c的N-周期序列的个数，并给出了线性复杂度的数学期望E〓的值以及k-错线性复杂度的数学期望E〓的一个有用的下界，这里p表示有限域GF的特征。

In the following two cases:(1)gcd = 1 ;(2)N = p~v, where p denotes thecharacteristic of the finite field GF, by using the factorization of the polynomialin finite field, the counting function N_(N,0) is showed, the expected value of thelinear complexity E_(N,0) is determined, and a useful lower bound on the expectedvalue of the k-error linear complexity E_ is established.(7) The stability of the lattice structure for pseudorandom number sequences is putforward.

利用多项式的因式分解等简单工具，研究了任意有限域GF上，周期N与p互素以及N = p~v这两种情况下，计数函数NN，0 的值，即线性复杂度为c的N-周期序列的个数，并给出了线性复杂度的数学期望EN，0的值以及k-错线性复杂度的数学期望EN，k的一个有用的下界，这里p表示有限域GF的特征。

Aerial image can be computed fairly fast by this method compared with the rigorous solution of the Hopkins equation, especially for the situation of many sparse aerial points which is often met in OPC application.

一个双线性光学系统分解成为一组空间域卷积核，并通过对版图的空间域卷积来计算空间光强。与采用Hopkins公式的SPLAT相比，这种方法能快速地计算空间光强。尤其对于大面积计算显得更为有效。

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。