This book provides the basic theory, techniques, and algorithms of modern cryptography that are applicable to network and cyberspace security. It consists of the following nine main chapters: Chapter 1 provides the basic concepts and ideas of cyberspace and cyberspace security, Chapters 2 and 3 provide an introduction to mathematical and computational preliminaries, respectively. Chapters 4 discusses the basic ideas and system of secret-key cryptography, whereas Chapters 5, 6, and 7 discuss the basic ideas and systems of public-key cryptography based on integer factorization, discrete logarithms, and elliptic curves, respectively. Quantum-safe cryptography is presented in Chapter 8 and offensive cryptography, particularly cryptovirology, is covered in Chapter 9. This book can be used as a secondary text for final-year undergraduate students and first-year postgraduate students for courses in Computer, Network, and Cyberspace Security. Researchers and practitioners working in cyberspace security and network security will also find this book useful as a reference.

There are many surprising connections between the theory of numbers, which is one of the oldest branches of mathematics, and computing and information theory. Number theory has important applications in computer organization and security, coding and cryptography, random number generation, hash functions, and graphics. Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. This book takes the reader from elementary number theory, via algorithmic number theory, to applied number theory in computer science. It introduces basic concepts, results, and methods, and discusses their applications in the design of hardware and software, cryptography, and security. It is aimed at undergraduates in computing and information technology, but will also be valuable to mathematics students interested in applications. In this 2nd edition full proofs of many theorems are added and some corrections are made.

RSA is a public-key cryptographic system, and is the most famous and widely-used cryptographic system in today's digital world. Cryptanalytic Attacks on RSA, a professional book, covers almost all known cryptanalytic attacks and defenses of the RSA cryptographic system and its variants. Since RSA depends heavily on computational complexity theory and number theory, background information on complexity theory and number theory is presented first, followed by an account of the RSA cryptographic system and its variants.

This book is also suitable as a secondary text for advanced-level students in computer science and mathematics.

This book provides a comprehensive introduction to advanced topics in the computational and algorithmic aspects of number theory, focusing on applications in cryptography. Readers will learn to develop fast algorithms, including quantum algorithms, to solve various classic and modern number theoretic problems. Key problems include prime number generation, primality testing, integer factorization, discrete logarithms, elliptic curve arithmetic, conjecture and numerical verification. The author discusses quantum algorithms for solving the Integer Factorization Problem (IFP), the Discrete Logarithm Problem (DLP), and the Elliptic Curve Discrete Logarithm Problem (ECDLP) and for attacking IFP, DLP and ECDLP based cryptographic systems. Chapters also cover various other quantum algorithms for Pell's equation, principal ideal, unit group, class group, Gauss sums, prime counting function, Riemann's hypothesis and the BSD conjecture. Quantum Computational Number Theory is self-contained and intended to be used either as a graduate text in computing, communications and mathematics, or as a basic reference in the related fields. Number theorists, cryptographers and professionals working in quantum computing, cryptography and network security will find this book a valuable asset.

The cryptosystems based on the Integer Factorization Problem (IFP), the Discrete Logarithm Problem (DLP) and the Elliptic Curve Discrete Logarithm Problem (ECDLP) are essentially the only three types of practical public-key cryptosystems in use. The security of these cryptosystems relies heavily on these three infeasible problems, as no polynomial-time algorithms exist for them so far. However, polynomial-time quantum algorithms for IFP, DLP and ECDLP do exist, provided that a practical quantum computer exists.

"Quantum Attacks on Public-Key Cryptosystems" presemts almost allknown quantum computing based attacks on public-key cryptosystems, with an emphasis on quantum algorithms for IFP, DLP, and ECDLP. It also discusses some quantum resistant cryptosystems to replace the IFP, DLP and ECDLP based cryptosystems.

This book is intended to be used either as a graduate text in computing, communications and mathematics, or as a basic reference in the field.

The Primality Testing Problem (PTP) has now proved to be solvable in deterministic polynomial-time (P) by the AKS (Agrawal-Kayal-Saxena) algorithm, whereas the Integer Factorization Problem (IFP) still remains unsolvable in (P). There is still no polynomial-time algorithm for IFP. Many practical public-key cryptosystems and protocols such as RSA (Rivest-Shamir-Adleman) rely their security on computational intractability of IFP. Primality Testing and Integer Factorization in Public Key Cryptography, Second Edition, provides a survey of recent progress in primality testing and integer factorization, with implications to factoring based public key cryptography. Notable new features are the comparison of Rabin-Miller probabilistic test in RP, Atkin-Morain elliptic curve test in ZPP and AKS deterministic test. This volume is designed for advanced level students in computer science and mathematics, and as a secondary text or reference book; suitable for practitioners and researchers in industry. First edition was very positively reviewed by Prof Samuel Wagstaff at Purdue University in AMS Mathematical Reviews (See MR2028480 2004j: 11148), and by Professor J.T. Ayuso of University of Simon Bolivar in the European Mathematical Societya (TM)s review journal Zentralblatt fA1/4r Mathematik (see Zbl 1048.11103).

This book provides a comprehensive introduction to advanced topics in the computational and algorithmic aspects of number theory, focusing on applications in cryptography. Readers will learn to develop fast algorithms, including quantum algorithms, to solve various classic and modern number theoretic problems. Key problems include prime number generation, primality testing, integer factorization, discrete logarithms, elliptic curve arithmetic, conjecture and numerical verification. The author discusses quantum algorithms for solving the Integer Factorization Problem (IFP), the Discrete Logarithm Problem (DLP), and the Elliptic Curve Discrete Logarithm Problem (ECDLP) and for attacking IFP, DLP and ECDLP based cryptographic systems. Chapters also cover various other quantum algorithms for Pell's equation, principal ideal, unit group, class group, Gauss sums, prime counting function, Riemann's hypothesis and the BSD conjecture. Quantum Computational Number Theory is self-contained and intended to be used either as a graduate text in computing, communications and mathematics, or as a basic reference in the related fields. Number theorists, cryptographers and professionals working in quantum computing, cryptography and network security will find this book a valuable asset.

The cryptosystems based on the Integer Factorization Problem (IFP), the Discrete Logarithm Problem (DLP) and the Elliptic Curve Discrete Logarithm Problem (ECDLP) are essentially the only three types of practical public-key cryptosystems in use. The security of these cryptosystems relies heavily on these three infeasible problems, as no polynomial-time algorithms exist for them so far. However, polynomial-time quantum algorithms for IFP, DLP and ECDLP do exist, provided that a practical quantum computer exists. Quantum Attacks on Public-Key Cryptosystems presemts almost all known quantum computing based attacks on public-key cryptosystems, with an emphasis on quantum algorithms for IFP, DLP, and ECDLP. It also discusses some quantum resistant cryptosystems to replace the IFP, DLP and ECDLP based cryptosystems. This book is intended to be used either as a graduate text in computing, communications and mathematics, or as a basic reference in the field.

This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information technology, including computer systems design, cryptography and network security. In this second edition proofs of many theorems have been provided, further additions and corrections were made.

Intended for advanced level students in computer science and mathematics, this key text, now in a brand new edition, provides a survey of recent progress in primality testing and integer factorization, with implications for factoring based public key cryptography. For this updated and revised edition, notable new features include a comparison of the Rabin-Miller probabilistic test in RP, the Atkin-Morain elliptic curve test in ZPP and the AKS deterministic test.

RSA is a public-key cryptographic system, and is the most famous and widely-used cryptographic system in today's digital world. Cryptanalytic Attacks on RSA, a professional book, covers almost all known cryptanalytic attacks and defenses of the RSA cryptographic system and its variants. Since RSA depends heavily on computational complexity theory and number theory, background information on complexity theory and number theory is presented first, followed by an account of the RSA cryptographic system and its variants.

This book is also suitable as a secondary text for advanced-level students in computer science and mathematics.