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Open Access 2022 | Open Access | Book

Modern Cryptography Volume 1

A Classical Introduction to Informational and Mathematical Principle

Author: Prof. Zhiyong Zheng

Publisher: Springer Singapore

Book Series : Financial Mathematics and Fintech

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Table of Contents

About this book

This open access book systematically explores the statistical characteristics of cryptographic systems, the computational complexity theory of cryptographic algorithms and the mathematical principles behind various encryption and decryption algorithms. The theory stems from technology. Based on Shannon's information theory, this book systematically introduces the information theory, statistical characteristics and computational complexity theory of public key cryptography, focusing on the three main algorithms of public key cryptography, RSA, discrete logarithm and elliptic curve cryptosystem. It aims to indicate what it is and why it is. It systematically simplifies and combs the theory and technology of lattice cryptography, which is the greatest feature of this book.
It requires a good knowledge in algebra, number theory and probability statistics for readers to read this book. The senior students majoring in mathematics, compulsory for cryptography and science and engineering postgraduates will find this book helpful. It can also be used as the main reference book for researchers in cryptography and cryptographic engineering areas.

Frontmatter

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Chapter 1. Preparatory Knowledge

Abstract

Modern cryptography and information theory is a branch of mathematics which develops rapidly. Almost all mathematical knowledge, such as algebra, geometry, analysis, probability and statistics, has very important applications in information theory.

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Chapter 2. The Basis of Code Theory

Abstract

The channel of information transmission is called channel for short. The commonly used channels include cable, optical fiber, medium of radio wave transmission and carrier line, etc., and also include tape, optical disk, etc. The channel constitutes the physical conditions for social information to interact across space and time. In addition, a piece of social information, such as various language information, picture information, data information and so on, should be exchanged across time and space, information coding is the basic technical means. What is information coding? In short, it is the process of digitizing all kinds of social information.

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Chapter 3. Shannon Theory

Abstract

According to Shannon, a message x is a random event. Let p(x) be the probability of occurrence of event x. If \(p(x)=0\), this event does not occur; If \(p(x)=1\), this event must occur. When \(p(x) = 0\) or \(p(x) = 1\), information x can be called trivial information or spam information. Therefore, the real mathematical significance of information x lies in its uncertainty, that is \(0<p(x)<1\). Quantitative research on the uncertainty of nontrivial information constitutes all the starting point of Shannon’s theory, this starting point is now called information quantity or information entropy, or entropy for short. Shannon and his colleagues at Bell laboratory considered “bit” as the basic quantitative unit of information. What is “bit”? We can simply understand it as the number of bits in the binary system. However, according to Shannon, the binary system with n digits can express up to \(2^{n}\) numbers. From the point of view of probability and statistics, the probability of occurrence of these \(2^{n}\) numbers is \(\frac{1}{2^{n}}\). Therefore, a bit is the amount of information contained in event x with probability \(\frac{1}{2}\). Taking this as the starting point, Shannon defined the self information I(x) contained in an information x as

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Chapter 4. Cryptosystem and Authentication System

Abstract

In 1949, Shannon published a famous paper entitled “communication theory of secure systems” in the technical bulletin of Bell laboratory. Based on the mathematical theory of information established by him in 1948 (see Chap. 3), this paper makes a comprehensive discussion on the problem of secure communication and establishes the mathematical theory of secure communication system. It has a great impact on the later development of cryptography. It is generally believed that Shannon transformed cryptography from art (creative ways and methods) to science, so he is also known as the father of modern cryptography. The main purpose of this chapter is to introduce Shannon’s important ideas and results in cryptography theory, which is the cornerstone of the whole modern cryptography.

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Chapter 5. Prime Test

Abstract

In the RSA algorithm in the previous chapter, we see that the decomposition of large prime factors constitutes the basis of RSA cryptosystem security. Theoretically, this security should not be questioned, because there is only the definition of prime in mathematics, and there is no general method to detect prime. The main purpose of this chapter is to introduce some basic prime test methods, including Fermat test, Euler test, Monte Carlo method, continued fraction method, etc. understanding the content of this chapter requires some special number theory knowledge.

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Chapter 6. Elliptic Curve

Abstract

In 1985, mathematician v. Miller introduced elliptic curve into cryptography for the first time. In 1987, mathematician N. Koblitz further improved and perfected Miller’s work and formed the famous elliptic curve public key cryptosystem.

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Chapter 7. Lattice-Based Cryptography

Let \(\mathbb {R}^n\) be an n-dimensional Euclidean space and \(x=(x_1, x_2, \ldots , x_n)\in \mathbb {R}^n\) be an n-dimensional vector, x can be a row vector or a column vector, depending on the situation. If \(x\in \mathbb {Z}^n\), then x is called a integral point. \(\mathbb {R}^{m\times n}\) is all \(m\times n\)-dimensional matrices on \(\mathbb {R}\). \(x=(x_1, x_2, \ldots , x_n)\in \mathbb {R}^n\), \(y=(y_1, y_2, \ldots , y_n)\in \mathbb {R}^n\), define the inner product of x and y as.

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Backmatter

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Title: Modern Cryptography Volume 1
Author: Prof. Zhiyong Zheng
Publisher: Springer Singapore
Electronic ISBN: 978-981-19-0920-7
Print ISBN: 978-981-19-0919-1
DOI: https://doi.org/10.1007/978-981-19-0920-7