Mathematical Foundations of Public Key Cryptography

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 IntegersThe Concept of Divisibility The Greatest Common Divisor and The Least Common Multiple The Euclidean Algorithm Solving Linear Diophantine Equations Prime Factorization of IntegersCongruences Residue Classes and Systems of ResiduesEuler’s TheoremWilson’s Theorem Congruence Equations Basic Concepts of Congruences of High Degrees Linear Congruences Systems of Linear Congruence Equations and the Chinese Remainder TheoremGeneral Congruence Equations Quadratic Residues The Legendre Symbol and the Jacobi Symbol Exponents and Primitive RootsExponents and Their PropertiesPrimitive 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 DistributionProof of the Euler Product FormulaProof of a Weaker Version of the Prime Number TheoremEquivalent 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 MapsAlgebraic OperationsHomomorphisms and Isomorphisms between Sets with Operations Equivalence Relations and Partitions Group Theory Definitions Cyclic Groups Subgroups and Cosets Fundamental Homomorphism TheoremConcrete Examples of Finite Groups Rings and FieldsDefinition of a Ring Integral Domains, Fields, and Division Rings Subrings, Ideals, and Ring HomomorphismsChinese 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 ProblemLattice Basis Reduction Algorithm Applications of LLL Algorithm References Further Reading Index

Dr. Xiaoyun Wang is a professor at the Institute of Advanced Study, Tsinghua University, China. She is also the director of the Center for Cryptology Study at Tsinghua University and an adjunct professor in the Key Lab of Cryptographic Technology and Information Security at Shandong University, China.Dr. Guangwu Xu is an associate professor in the Department of Electrical Engineering and Computer Science, University of Wisconsin-Milwaukee. Dr. Xu’s research concerns the efficiency, security, and reliability of information processing. He is interested in the fundamental problems of these aspects. Dr. Mingqiang Wang earned his PhD degree in 2004 from Shandong University, China where he serves as a professor now. Dr. Wang is a member of the Chinese Association for Cryptologic Research, his research focuses on number theory and analysis and design of public key algorithms. Dr. Xianmeng Meng earned herbachelor's and master's degrees from Jilin University in 1993 and 1996 respectively, then graduated with a PhD degree from Shandong University, China. She is a member of the Chinese Association for Cryptologic Research and is currently a professor in Shandong University of Finance and Economics, her main research interest is number theory and cryptography.