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# 数学代写|密码学代写Cryptography Theory代考|Quantum Computers and Quantum Distributed Key Networks

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## 数学代写|密码学Cryptography Theory代考|Quantum Computers and Quantum Distributed Key Networks

Traditional computers operate using bits that are either 0 or 1 . In the old days, these 0 s and 1 s were represented by vacuum tubes that were either $\mathrm{OFF}$ or $\mathrm{ON}$. The space needed to store bits was reduced dramatically over the decades and the tubes are long gone. But a further reduction has recently been made. In quantum computers it is actually quantum particles that are used to represent the bits. However, there is a fundamental difference. It is not just a matter of a smaller size. Quantum bits, or qubits (pronounced “cue bits”) for short, can be 0,1 , or both. A description of how quantum computers work is well outside the scope of this book. What’s relevant is that these machines can solve some problems that traditional computers cannot and can solve other problems far faster. For example, there’s no known polynomial time algorithm for factoring, using a traditional computer, but there is one for a quantum computer. It dates back to 1994 and is known as Shor’s algorithm, after Peter Shor who was employed by Bell Labs at the time. ${ }^{23}$ Shor also found a polynomial time algorithm for solving the discrete log problem on a quantum computer. As a consequence, RSA, Diffie-Hellman, and elliptic curve cryptography are all vulnerable.

It’s not just public key systems that are at risk. Grover’s algorithm, discovered by Lov Grover, an Indian-American computer scientist, in 1996, ${ }^{24}$ can be used to reduce the number of trials needed to brute-force a symmetric block cipher with an $n$ bit key from $2^n$ to $2^{n / 2}$ on a quantum computer. ${ }^{25}$ The ability of a qubit to be both 0 and 1 allows many keys to be tested simultaneously. The October 23, 2019 issue of Nature contained a paper by 77 authors (representing Google). The abstract include a dramatic summary of the power of a quantum computer with 53 qubits:
Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times-our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy for this specific computational task, heralding a much anticipated computing paradigm. ${ }^{26}$
IBM, a competitor in the quantum computer development race, objected to this claim, saying that the time on a state-of-the-art classical supercomputer is 2.5 days, not 10,000 years. ${ }^{27}$ As of May 2020, IBM has 18 quantum computers, Honeywell has 6 , and Google has 5.28

One way to protect communications against such new machines (as well as improved versions, yet to be, that will make these look like toys) is by setting up a quantum key distribution network, as described earlier in this chapter. Another is to replace current algorithms with ones believed to be able to resist quantum computer attacks. The next two sections detail how NSA and NIST are slowly prodding people in this direction.

## 数学代写|密码学Cryptography Theory代考|NSA Weighs In

In Chapter 20, some algorithms recommended by the National Security Agency as part of their “Suite B Cryptography” were detailed. In light of the threat of quantum computers, NSA introduced the “Commercial National Security Algorithm Suite (CNSA Suite)” on August 19, 2015. These algorithms were only intended as a stopgap measure. The agency promised, “IAD Information Assurance Directorate will initiate a transition to quantum resistant algorithms in the not too distant future.” ${ }^{29}$ NSA also gave advice for those who were a bit behind and had not upgraded to Suite B:
Until this new quantum resistant algorithms suite is developed and products are available implementing the quantum resistant suite, we will rely on current algorithms. For those partners and vendors that have not yet made the transition to Suite B elliptic curve algorithms, we recommend not making a significant expenditure to do so at this point but instead to prepare for the upcoming quantum resistant algorithm transition. ${ }^{30}$
The CNSA Suite did not contain any new algorithms. The list had the old popular schemes like AES, Elliptic Curve schemes, SHA, Diffie-Hellman, and RSA. That is RSA was placed in higher esteem than in Suite B and DSA was dropped. The main difference in the retained algorithms was that the key sizes were much larger. For example, for Diffie-Hellman key exchange, it was “Minimum 3072-bit modulus to protect up to TOP SECRET.”
The other newsworthy update was expressed as follows:
Unfortunately, the growth of elliptic curve use has bumped up against the fact of continued progress in the research on quantum computing, which has made it clear that elliptic curve cryptography is not the long term solution many once hoped it would be. Thus, we have been obligated to update our strategy. ${ }^{32}$
These lines led to much speculation, a summary of which was presented in a paper by Neal Koblitz, a co-discoverer of elliptic curve cryptography, and Alfred J. Menezes. ${ }^{33}$ In an email to me, Koblitz noted, “It’s interesting that one of the leading contenders for “post-quantum cryptography” is based on elliptic curves, but in a totally different way from ECC. This is the “isogeny-based” approach of Jao and others.” ${ }^{34}$

## 数学代写|密码学Cryptography Theory代考|NSA Weighs In

NSA 还为那些有点落后且没有升级到 Suite B 的人提供了建议：

CNSA Suite 不包含任何新算法。该列表包含旧的流行方案，如 AES、椭圆曲线方案、SHA、Diffie-Hellman 和 RSA。也就是说，RSA 比 Suite B 受到更高的重视，而 DSA 被丢弃了。保留算法的主要区别在于密钥大小要大得多。例如，对于 Diffie-Hellman 密钥交换，它是“最小 3072 位模数以保护最高机密”。

Koblitz 在给我的一封电子邮件中指出，“有趣的是，‘后量子密码学’的主要竞争者之一基于椭圆曲线，但与 ECC 的方式完全不同。这就是 Jao 等人的“基于等基因”的方法。”

## 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。