Librería Portfolio

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