Mathematical foundations of public key cryptography

Content: 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