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# Primality Testing and Integer Factorization in Public-Key Cryptography

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Januar 2004

## Beschreibung

### Beschreibung

Primality Testing and Integer Factorization in Public-Key Cryptography introduces various algorithms for primality testing and integer factorization, with their applications in public-key cryptography and information security. More specifically, this book explores basic concepts and results in number theory in Chapter 1. Chapter 2 discusses various algorithms for primality testing and prime number generation, with an emphasis on the Miller-Rabin probabilistic test, the Goldwasser-Kilian and Atkin-Morain elliptic curve tests, and the Agrawal-Kayal-Saxena deterministic test for primality. Chapter 3 introduces various algorithms, particularly the Elliptic Curve Method (ECM), the Quadratic Sieve (QS) and the Number Field Sieve (NFS) for integer factorization. This chapter also discusses some other computational problems that are related to factoring, such as the square root problem, the discrete logarithm problem and the quadratic residuosity problem.

### Inhaltsverzeichnis

Preface to the Second Edition.
Preface to the First Edition.
Number-Theoretic Preliminaries.
Problems in Number Theory. Divisibility Properties. Euclid's Algorithm and Continued Fractions. Arithmetic Functions. Linear Congruences. Quadratic Congruences. Primitive Roots and Power Residues. Arithmetic of Elliptic Curves. Chapter Notes and Further Reading.
Primality Testing and Prime Generation.
Computing with Numbers and Curves. Riemann Zeta and Dirichlet L Functions. Rigorous Primality Tests. Compositeness and Pseudoprimality Tests. Lucas Pseudoprimality Test. Elliptic Curve Primality Tests. Superpolynomial-Time Tests. Polynomial-Time Tests. Primality Tests for Special Numbers. Prime Number Generation. Chapter Notes and Further Reading.
Integer Factorization and Discrete Logarithms.
Introduction. Simple Factoring Methods. Elliptic Curve Method (ECM). General Factoring Congruence. Continued FRACtion Method (CFRAC). Quadratic Sieve (QS). Number Field Sieve (NFS). Quantum Factoring Algorithm. Discrete Logarithms. kth Roots. Elliptic Curve Discrete Logarithms. Chapter Notes and Further Reading.
Number-Theoretic Cryptography.
Public-Key Cryptography. RSA Cryptosystem. Rabin Cryptography. Quadratic Residuosity Cryptography. Discrete Logarithm Cryptography. Elliptic Curve Cryptography. Zero-Knowledge Techniques. Deniable Authentication. Non-Factoring Based Cryptography. Chapter Notes and Further Reading.
Bibliography.
Index.