Information Security Mathematical Foundation

Author: Li Jiguo
Publisher: Wuhan University Press
Book Description:
This book comprehensively and systematically introduces fundamental theories in number theory, algebra, combinatorial mathematics, and some practical algorithms used in cryptography. It covers mathematical knowledge such as divisibility, congruences, quadratic congruences and quadratic residues, primitive roots, groups, rings, finite fields, lattices and their applications, elliptic curves, and combinatorial mathematics, as well as practical algorithms like primality testing, factorization, and discrete logarithm computation. The book consists of 13 chapters and includes a list of main references at the end. It can serve as a textbook for undergraduate and graduate students in information security, computer science and technology, communication engineering, mathematics and applied mathematics, and is also a valuable reference for researchers and engineers working in information security, cryptography, and other information technologies.
Table of Contents:
Chapter 1 Divisibility
1.1 Basic Properties of Divisibility and the Remainder Theorem
1.2 Greatest Common Divisor and Least Common Multiple
1.3 Fundamental Theorem of Arithmetic
1.4 Experiments
1.5 Exercises
Chapter 2 Congruences
2.1 Definition and Basic Properties of Congruences
2.2 Residue Classes and Residue Systems
2.3 Several Famous Theorems
2.4 RSA Public Key Cryptography System
2.4.1 Key Generation
2.4.2 RSA System
2.4.3 RSA Security
2.4.4 Choice of RSA Parameters
2.5 Congruences
2.6 Linear Congruences
2.7 Chinese Remainder Theorem
2.8 Methods and Number of Solutions for Higher-Degree Congruences
2.9 Solving Higher-Degree Congruences with Prime Moduli
2.10 Experiments
2.11 Exercises
Chapter 3 Quadratic Congruences and Quadratic Residues
3.1 Concepts of Quadratic Congruences and Quadratic Residues
3.2 Quadratic Residues and Non-Residues Modulo Odd Primes
3.3 Legendre Symbol
3.4 Jacobi Symbol
3.5 Modulo P Square Roots
3.6 Case of Composite Moduli
3.7 Experiments
3.8 Exercises
Chapter 4 Primitive Roots
4.1 Exponent and Its Basic Properties
4.2 Primitive Roots and Their Computation
4.3 Indicators and n-Terminal Residues
4.4 Experiments
4.5 Exercises
Chapter 5 Groups
5.1 Preliminary Knowledge
5.1.1 Concept of Binary Operations
5.1.2 Properties of Binary Operations
5.1.3 Definition of Algebraic Systems
5.2 Definition and Properties of Groups
5.2.1 Definition of Groups
5.2.2 Order of Elements in a Group
5.2.3 Subgroups and Subgroup Determination
5.3 Homomorphisms and Isomorphisms
5.3.1 Definitions of Homomorphisms and Isomorphisms
5.3.2 Properties of Homomorphisms
5.4 Cyclic Groups and Permutation Groups
5.5 Applications of Groups
5.6 Exercises
Chapter 6 Rings
6.1 Definition and Properties of Rings
6.2 Integral Domains and Fields
6.3 Applications of Rings
6.3.1 Idempotent Elements in Non-Negative Integer Rings and Their Properties
6.3.2 Implementation of Encryption Algorithms Based on Idempotent Elements
6.3.3 Examples of Encryption Algorithms
6.4 Exercises
Chapter 7 Finite Field Theory
7.1 Field Extensions
7.2 Basic Concepts and Properties of Finite Fields
7.3 Minimal Polynomials and Primitive Polynomials
7.4 Polynomial Periodicity
Chapter 8 Lattices and Their Applications
Chapter 9 Elliptic Curves
Chapter 10 Combinatorics
Chapter 11 Primality Testing
Chapter 12 Factorization
Chapter 13 Discrete Logarithm Computation
Appendix
References