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

## 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.

# 密码学代考

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

. crypgraphy theory . crypgraphy theory

$$y^2=x^3+A x+B$$

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