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# 数学代写|密码学代写Cryptography Theory代考|CSE546 GIMPS

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## 数学代写|密码学Cryptography Theory代考|Pollard’s $p-1$ Algorithm

Recall Fermat’s Little Theorem:
If $p$ is a prime and $a$ is not a multiple of $p$, then $a^{p-1}=1(\bmod p)$.
So, if $m$ is a multiple of $p-1$, then $a^m=1(\bmod p)$. In other words, $p$ divides $a^m-1$. So, $\operatorname{gcd}\left(a^m-1, n\right)$, where $a^m-1$ is first reduced $\bmod n$, might reveal a factorization of $n$, because $p$ divides both $a^m-1$ and $n$.

But how can we find a multiple $m>1$ of $p-1$ ? Well, we hope that $p-1$ is $B$-smooth (i.e., all prime factors of $p-1$ are less than $B)^{22}$ for some small $B$. We let $m$ be defined as the product of all primes $q$ less than $B$. Just as a capital sigma denotes a summation, a capital pi denotes a product. We have
$$m=\prod_{q \leq B} q \quad q \text { prime }$$

## 数学代写|密码学Cryptography Theory代考|GIMPS

Some numbers are easier to test for primality than others. Mersenne numbers, which have the form $2^n-1$, are currently the easiest. For values of $n$, such as $n=2$ or 3 , that yield primes, we call the numbers Mersenne primes, after the French mathematician Marin Mersenne (1588-1648). The ease of testing such numbers is why the top 8 largest known primes are all Mersenne primes. Another advantage numbers of this form have, when it comes to a chance of making it on the top 10 list, is that anyone with a computer may download a program that allows him or her to join in the testing. The program, Great Internet Mersenne Prime Search (GIMPS, for short), allows people to donate otherwise idle time on their personal computers to testing numbers of the form $2^n-1$ for primality. In the top 10 list, the number referenced as Mersenne 47 is known to be the 47th Mersenne prime. By contrast, the number referenced as Mersenne 48? is known to be a Mersenne prime, but it might not be the 48 th such number. That is, mathematicians have yet to prove that there is no Mersenne prime between Mersenne 47 and this number.

There is one prime on the current top 10 list that is not a Mersenne prime and wasn’t discovered through GIMPS. It was credited to “Seventeen or Bust,” which is another program that anyone may download to help search for primes taking a special form, in this case the form $k \cdot 2^n+1$ for certain values of $k{ }^{22}$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。