乘法定理

In this thesis, sound wave path equation is derived from Fermat's theorem and the calculus of variations, and then the temperature field is reconstructed by the combination least-square method and the modified sound wave path.

本文通过求解由费马定理和变积分原理确定的声波路径方程，结合最小二乘法实现了声波路径的修正。

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。

Oseledec\'s Multiplicative Ergodic Theorem does not only solve the existence of Lyapunov exponents, but also show much more information of the dynamical structure.

Oseledec的乘法遍历定理解决了Lyapunov指数的存在性问题，并对动力系统的动力学结构给出了更多的信息，它现在已成为动力系统理论的最基本定理之一。

The author mainly realized the method of generating the large random numbers and judging the prime number using Lehmann means,then three pairs of secure key with different lengths (256 bits,512 bits,1024 bits) was generated to meet the demand of different security grade. In encryption module,modular multiplication arithmetic of large number was realized adopting addition chaining. In decryption module,extended Euclid was used to get inverse,and the Chinese remainder theorem was used to realize decryption,which increased the speed of decryption greatly.

作者主要实现了大随机数的生成、采用整除小素数和勒曼测试法判定素数，可以生成三种不同长度的密钥对(256位、512位、1024位)来满足不同的加密安全等级的需求；加密模块中，选用了二进制序列的乘法方法，实现了大数模幂乘运算；解密模块中，引用了扩展的欧几里德法求逆元，并采用中国剩余定理实现解密过程，大大提高了解密速度。

Then several structural theorems of regular power rings and uniform power rings are established respectively and the uniqueness of the multiplication of a regular power ring is proved.

建立了正则幂环与一致幂环的结构定理，并证明了正则幂环的乘法运算的唯一性。

In this paper, introduces firstly a concept of polynomial coefficient matrix, and then the polynomial multiplication is masterly changed into the computing of matrix multiplication. Then a theorem is received. Throughout the process, the steps are very clear and the calculating is pretty simple.

该文首先引进了一个多项式系数矩阵的概念，然后巧妙地将多项式的乘法转变为矩阵乘法的运算，得到了一个定理，步骤清晰，计算简单。

And the Montgomery method to realize point multiplication is also efficient by using projective coordinates to reduce the multiplication times and only one time of inversion been needed.

利用Fermatt小定理求逆，根据有限域上元素的乘法满足结合律和交换律，通过重新安排乘法和平方的顺序，可以减少乘法和平方的次数，比按照定义直接实现速度提高了6倍。

Secondly, the properties of output sequences of stop-and-go clocked combiners are discussed. It is proved that the output sequences are stationary and obey strong law of large numbers and central limit theorem. Using the probabilistic model properly built, we analyze the rate of coincidence between the output sequences of two kinds of generators with basic operations and corresponding LFSRs sequences. All the computational formulae are derived. A recursive algorithm for efficiently computing the poster probability of partial input strings and corresponding joint probability is introduced. Moreover, divide-and-conquer attacks on this kind of generators which reconstruct the initial states of the input stop-and-go generators individually are proposed.

其次，分析了一般钟控停走组合生成器输出序列的概率分布性质，证明了生成器的输出序列是严平稳的，且服从强大数定律和中心极限定理；考察了钟控加法型组合生成器和钟控乘法型组合生成器的输出序列和相应的常规钟控下线性移位寄存器序列之间的符合率，给出了符合率的具体计算公式；给出了组合器输出序列段和输入序列段之间的联合分布以及部分输入序列段的后验概率的有效递归算法，进一步提出了利用后验概率对钟控组合生成器进行分别征服相关攻击的方法。

Neither the killing of Mr Zarqawi nor any breakthrough on the political front will stop the insurgency and the fratricidal murders in their tracks.

在对危险的南部地区访问时，他斥责什叶派民兵领导人对中央集权的挑衅行为。

In fact,I've got him on the satellite mobile right now.

实际上 我们已接通卫星可视电话了