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数学代写|密码学代写cryptography theory代考|Elliptic Curve Cryptography

This particular algorithm may be the most mathematically challenging that you will encounter in this book, or in any other introductory cryptography book. If you feel uncertain as to your mathematical acumen, you may wish to review Chaps. 4 and 5 (with particular attention to 5). Throughout the chapter, there are brief reminders as to key mathematical concepts to help you follow along. If your goal is a career related to cryptography, then you will, at some point, need to master this material. However, for those readers attempting to get a general overview of cryptography, whose primary focus is on computer/network security, then it is perfectly acceptable if you finish this chapter with just a broad overview of elliptic curve cryptography. Since this topic is often difficult for many readers, a chapter (albeit a short one) has been devoted to just this topic. Furthermore, in some cases, key concepts are explained more than once with slightly different wording to try and aid your understanding.

The reason why elliptic curve cryptography (commonly termed ECC) is more difficult for many people to learn is that fewer people have any prior exposure to the underlying mathematics. When one compares ECC to RSA, this difference becomes quite clear. Most people were exposed to prime numbers, factoring numbers, raising a number to a certain power, and basic arithmetic in primary and secondary school. But far fewer people are exposed to elliptic curves and discrete logarithms in school.

数学代写|密码学代写cryptography theory代考|General Overview

Elliptic curves have been studied, apart from cryptographic applications, for well over a century. As with other asymmetric algorithms, the mathematics has been a part of number theory and algebra, long before being applied to cryptography. As you saw in Chap. 10, many asymmetric algorithms depend on algebraic groups. There are multiple means to form finite groups. Elliptic curves can be used to form groups, and thus are appropriate for cryptographic purposes. There are two types of elliptic curve groups. The two most common (and the ones used in cryptography) are elliptic curve groups based on $\mathrm{F}_{\mathrm{p}}$ where $\mathrm{p}$ is prime and those based on $\mathrm{F}^{\mathrm{m}}$ (Rabah 2006). F, as you will see in this chapter, is the field being used. F is used because we are describing a field. Elliptic curve cryptography is an approach to public-key cryptography, based on elliptic curves over finite fields.

Remember that a field is an algebraic system consisting of a set, an identity element for each operation, two operations, and their respective inverse operations. A finite field, also called a Galois field, is a field with a finite number elements. That number is called the order of the field. Elliptic curves used for cryptographic purposes were first described in 1985 by Victor Miller (IBM) and Neil Koblitz (University of Washington). The security of elliptic curve cryptography is based on the fact that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is difficult to the point of being impractical to do (Rabah 2006).

Neal Koblitz is a mathematics professor at the University of Washington and a very well-known cryptographic researcher. In addition to his work on elliptic curve cryptography, he has published extensively in mathematics and cryptography. Victor Miller is a mathematician with the Institute for Defense Analysis in Princeton. He has worked on compression algorithms, combinatorics, and various subtopics in the field of cryptography.

First, we need to discuss what an elliptic curve is. An elliptic curve is the set of points that satisfy a specific mathematical equation. The equation for an elliptic curve looks something like this (Hankerson et al. 2006):
$$
y^2=x^3+A x+B
$$
You can see this equation graphed in Fig. 11.1.

数学代写|密码学代写cryptography theory代考|CISS3341

密码学代考

数学代写|密码学代写cryptography theory代考|椭圆曲线密码学

这个特殊的算法可能是你在本书中,或在任何其他介绍性密码学书籍中遇到的最具数学挑战性的算法。如果你对自己的数学能力感到不确定,你可能希望复习第4章和第5章(特别注意第5章)。整章都有关于关键数学概念的简短提醒,以帮助你理解。如果您的目标是从事与密码学相关的职业,那么在某种程度上,您需要掌握这些材料。然而,对于那些主要关注计算机/网络安全的试图获得密码学概览的读者来说,如果您在本章结束时只对椭圆曲线密码学进行概览,则完全可以接受。由于这个主题对许多读者来说通常很难,因此专门用了一章(尽管很短)来讨论这个主题。此外,在某些情况下,为了帮助理解,关键概念会用稍微不同的措辞解释不止一次


为什么椭圆曲线密码学(通常被称为ECC)对许多人来说比较困难,原因是很少有人事先接触过底层的数学。当人们比较ECC和RSA时,这种差异就变得非常明显。大多数人在小学和中学时都接触过质数、因式分解、一个数的某次方以及基本的算术。但是在学校接触到椭圆曲线和离散对数的人却少得多

数学代写|密码学代写cryptography theory代考|General Overview

. crypgraphy theory . crypgraphy theory


除了密码应用之外,椭圆曲线的研究已经有一个多世纪了。与其他非对称算法一样,在应用到密码学之前,数学一直是数论和代数的一部分。正如你在第10章中看到的,许多非对称算法依赖于代数群。形成有限群有多种方法。椭圆曲线可以用来形成组,因此适合于加密目的。椭圆曲线群有两种类型。两个最常见的(也是密码学中使用的)是基于$\mathrm{F}_{\mathrm{p}}$的椭圆曲线组(其中$\mathrm{p}$是素数)和基于$\mathrm{F}^{\mathrm{m}}$的椭圆曲线组(Rabah 2006)。F,正如你将在本章中看到的,是所使用的字段。使用F是因为我们在描述一个场。椭圆曲线密码学是一种基于有限域上椭圆曲线的公钥密码学方法


记住,字段是一个代数系统,由一个集合、每个操作的一个单位元、两个操作以及它们各自的逆操作组成。有限场,也称为伽罗瓦场,是一个具有有限数量元素的场。这个数字叫做场的顺序。1985年,Victor Miller (IBM)和Neil Koblitz(华盛顿大学)首次描述了用于加密目的的椭圆曲线。椭圆曲线密密学的安全性是基于这样一个事实:找到一个随机椭圆曲线元素相对于一个公开的基点的离散对数是非常困难的,以至于难以实现(Rabah, 2006)


尼尔·科布利茨(Neal Koblitz)是华盛顿大学的数学教授,也是一位非常著名的密码研究者。除了在椭圆曲线密码学方面的工作外,他还在数学和密码学方面发表了大量文章。维克多·米勒(Victor Miller)是普林斯顿国防分析研究所的数学家。他曾在密码学领域的压缩算法、组合学和各种子主题上工作


首先,我们需要讨论什么是椭圆曲线。椭圆曲线是满足特定数学方程的点的集合。椭圆曲线的方程是这样的(Hankerson et al. 2006):
$$
y^2=x^3+A x+B
$$
你可以在图11.1中看到这个方程

数学代写|密码学代写cryptography theory代考

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