Posted on Categories:Cryptography, 密码学, 数学代写

# 数学代写|密码学代写Cryptography Theory代考|Elliptic Curve Cryptography

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## 数学代写|密码学Cryptography Theory代考|Elliptic Curve Cryptography

Elliptic Curve Cryptography (ECC) is a phrase used to describe a suite of cryptographic primitives and protocols whose security is based on special versions of the discrete logarithm problem. Instead of using the numbers modulo $p$, ECC is based on different sets of numbers. These numbers are associated with mathematical objects called elliptic curves. There are rules for adding and computing multiples of these numbers, just as there are for numbers modulo $p$. We will not concern ourselves here with any of the details of elliptic curves or how to combine the points on such a curve.
ECC includes a number of variants of cryptographic primitives which were first designed for modular numbers. As well as variants of ElGamal encryption, these include an elliptic-curve-based variant of the Diffie-Hellman key agreement protocol (see Section 9.4.2), and an elliptic-curve-based variant of the Digital Signature Algorithm (see Section 7.3.6).

The advantage of switching from numbers modulo $p$ to points on an elliptic curve is that it is believed the discrete logarithm problem is much harder when applied to points on an elliptic curve. The important implication is that an equivalent security level can be obtained for shorter keys if we use elliptic-curve-based variants. We will show the approximate extent of this reduction in Section 5.4.

The many advantages of shorter keys, both in terms of key management and efficient computation (see Section 10.2), make elliptic-curve-based variants highly attractive for many application environments. ECC primitives are being increasingly adopted, especially in resource-constrained environments.

## 数学代写|密码学Cryptography Theory代考|Popularity of RSA

Historically there is no doubt RSA has been by far the most popular public-key cryptosystem. There are several possible reasons for this:

Maturity. RSA was one of the first public-key cryptosystems to be proposed and was the first to gain widespread recognition. Thus, in many senses, RSA is the brand leader.

Less message expansion. ElGamal involves message expansion by default, which makes its use potentially undesirable. The ‘textbook’ version of RSA has no message expansion, and RSA-OAEP has limited message expansion.

Marketing. The use of RSA was marketed from an early stage by a commercial company. Indeed, it was at one stage subject to patent in certain parts of the world. ElGamal has not had such successful commercial backing. However, ECC does, and there are a number of patents on ECC primitives.

## 数学代写|密码学Cryptography Theory代考|Elliptic Curve Cryptography

ECC包含许多最初为模数设计的密码原语变体。与ElGamal加密的变体一样，这些变体包括基于椭圆曲线的Diffie-Hellman密钥协议变体(参见第9.4.2节)和基于椭圆曲线的数字签名算法变体(参见第7.3.6节)。

## 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。