Mathematical Foundations of Public Key Cryptography

ISBN: 9781498702232 出版年：2015 页码：228 Wang, Xiaoyun Xu, Guangwu Wang, Mingqiang CRC Press

Divisibility of Integers The Concept of Divisibility The Greatest Common Divisor and The Least Common Multiple The Euclidean Algorithm Solving Linear Diophantine Equations Prime Factorization of Integers Congruences Residue Classes and Systems of Residues Euler's Theorem Wilson's Theorem Congruence Equations Basic Concepts of Congruences of High Degrees Linear Congruences Systems of Linear Congruence Equations and the Chinese Remainder Theorem General Congruence Equations Quadratic Residues The Legendre Symbol and the Jacobi Symbol Exponents and Primitive Roots Exponents and Their Properties Primitive Roots and Their Properties Indices, Construction of Reduced System of Residues Nth Power Residues Some Elementary Results for Prime Distribution Introduction to the Basic Properties of Primes and The Main Results of Prime Number Distribution Proof of the Euler Product Formula Proof of a Weaker Version of the Prime Number Theorem Equivalent Statements of the Prime Number Theorem Simple Continued Fractions Simple Continued Fractions and Their Basic Properties Simple Continued Fraction Representations of Real Numbers Application of Continued Fraction In Cryptography-Attack to RSA with Small Decryption Exponents Basic Concepts Maps Algebraic Operations Homomorphisms and Isomorphisms between Sets with Operations Equivalence Relations and Partitions Group Theory Definitions Cyclic Groups Subgroups and Cosets Fundamental Homomorphism Theorem Concrete Examples of Finite Groups Rings and Fields Definition of a Ring Integral Domains, Fields, and Division Rings Subrings, Ideals, and Ring Homomorphisms Chinese Remainder Theorem Euclidean Rings Finite Fields Field of Fractions Some Mathematical Problems in Public Key Cryptography Time Estimation and Complexity of Algorithms Integer Factorization Problem Primality Tests The RSA Problem and the Strong RSA Problem Quadratic Residues The Discrete Logarithm Problem Basics of Lattices Basic Concepts Shortest Vector Problem Lattice Basis Reduction Algorithm Applications of LLL Algorithm References Further Reading Index