Opis: Mathematical Foundations of Public Key Cryptography - Xianmeng Meng, Xiammeng Meng, Mingqiang Wang

In Mathematical Foundations of Public Key Cryptography, the authors integrate the results of more than 20 years of research and teaching experience to help students bridge the gap between math theory and crypto practice. The book provides a theoretical structure of fundamental number theory and algebra knowledge supporting public-key cryptography. Rather than simply combining number theory and modern algebra, this textbook features the interdisciplinary characteristics of cryptography-revealing the integrations of mathematical theories and public-key cryptographic applications. Incorporating the complexity theory of algorithms throughout, it introduces the basic number theoretic and algebraic algorithms and their complexities to provide a preliminary understanding of the applications of mathematical theories in cryptographic algorithms. Supplying a seamless integration of cryptography and mathematics, the book includes coverage of elementary number theory; algebraic structure and attributes of group, ring, and field; cryptography-related computing complexity and basic algorithms, as well as lattice and fundamental methods of lattice cryptanalysis. The text consists of 11 chapters. Basic theory and tools of elementary number theory, such as congruences, primitive roots, residue classes, and continued fractions, are covered in Chapters 1-6. The basic concepts of abstract algebra are introduced in Chapters 7-9, where three basic algebraic structures of groups, rings, and fields and their properties are explained. Chapter 10 is about computational complexities of several related mathematical algorithms, and hard problems such as integer factorization and discrete logarithm. Chapter 11 presents the basics of lattice theory and the lattice basis reduction algorithm-the LLL algorithm and its application in the cryptanalysis of the RSA algorithm. Containing a number of exercises on key algorithms, the book is suitable for use as a textbook for undergraduate students and first-year graduate students in information security programs. It is also an ideal reference book for cryptography professionals looking to master public-key cryptography.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

Szczegóły: Mathematical Foundations of Public Key Cryptography - Xianmeng Meng, Xiammeng Meng, Mingqiang Wang

Tytuł: Mathematical Foundations of Public Key Cryptography
Autor: Xianmeng Meng, Xiammeng Meng, Mingqiang Wang
Producent: Productivity Press Inc
ISBN: 9781498702232
Rok produkcji: 2015
Ilość stron: 236
Oprawa: Twarda

Recenzje: Mathematical Foundations of Public Key Cryptography - Xianmeng Meng, Xiammeng Meng, Mingqiang Wang