Prime Numbers A Computational Perspective

ISBN-10: 0387252827
ISBN-13: 9780387252827
Edition: 2nd 2005 (Revised)
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Description: Prime numbers beckon to the beginner, the basic notion of primality being accessible to a child. Yet, some of the simplest questions about primes have stumped humankind for millennia. In this book, the authors concentrate on the computational  More...

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Book details

Edition: 2nd
Copyright year: 2005
Publisher: Springer
Publication date: 8/4/2005
Binding: Hardcover
Pages: 597
Size: 6.25" wide x 9.50" long x 1.50" tall
Weight: 2.244
Language: English

Prime numbers beckon to the beginner, the basic notion of primality being accessible to a child. Yet, some of the simplest questions about primes have stumped humankind for millennia. In this book, the authors concentrate on the computational aspects of prime numbers, such as recognizing primes and discovering the fundamental prime factors of a given number. Over 100 explicit algorithms cast in detailed pseudocode are included in the book. Applications and theoretical digressions serve to illuminate, justify, and underscore the practical power of these algorithms. The 2 nd edition adds new material on primality and algorithms and updates all the numerical records, such as the largest prime, etc. It has been revised throughout. From the reviews of the first edition: "The exercises are a gold mine of interesting examples, pointers to the literature and potential research projects. Prime Numbers is a welcome addition to the literature of number theorycomprehensive, up-to-date and written with style. It will be useful to anyone interested in algorithms dealing with the arithmetic of the integers and related computational issues." American Scientist "Destined to become a definitive textbook conveying the most modern computational ideas about prime numbers and factoring, this book will stand as an excellent reference for this kind of computation, and thus be of interest to both educators and researchers. It is also a timely book, since primes and factoring have reached a certain vogue, partly because of cryptography. " LEnseignement Mathmatique "The book is an excellent resource for anyone who wants to understand these algorithms, learn how to implement them, and make them go fast. It's also a lot of fun to read! It's rare to say this of a math book, but open Prime Numbers to a random page and it's hard to put down. Crandall and Pomerance have written a terrific book." Bulletin of the AMS

Preface
Primes!
Problems and progress
Fundamental theorem and fundamental problem
Technological and algorithmic progress
The infinitude of primes
Asymptotic relations and order nomenclature
How primes are distributed
Celebrated conjectures and curiosities
Twin primes
Prime k-tuples and hypothesis H
The Goldbach conjecture
The convexity question
Prime-producing formulae
Primes of special form
Mersenne primes
Fermat numbers
Certain presumably rare primes
Analytic number theory
The Riemann zeta function
Computational successes
Dirichlet L-functions
Exponential sums
Smooth numbers
Exercises
Research problems
Number-Theoretical Tools
Modular arithmetic
Greatest common divisor and inverse
Powers
Chinese remainder theorem
Polynomial arithmetic
Greatest common divisor for polynomials
Finite fields
Squares and roots
Quadratic residues
Square roots
Finding polynomial roots
Representation by quadratic forms
Exercises
Research problems
Recognizing Primes and Composites
Trial division
Divisibility tests
Trial division
Practical considerations
Theoretical considerations
Sieving
Sieving to recognize primes
Eratosthenes pseudocode
Sieving to construct a factor table
Sieving to construct complete factorizations
Sieving to recognize smooth numbers
Sieving a polynomial
Theoretical considerations
Recognizing smooth numbers
Pseudoprimes
Fermat pseudoprimes
Carmichael numbers
Probable primes and witnesses
The least witness for n
Lucas pseudoprimes
Fibonacci and Lucas pseudoprimes
Grantham's Frobenius test
Implementing the Lucas and quadratic Frobenius tests
Theoretical considerations and stronger tests
The general Frobenius test
Counting primes
Combinatorial method
Analytic method
Exercises
Research problems
Primality Proving
The n - 1 test
The Lucas theorem and Pepin test
Partial factorization
Succinct certificates
The n + 1 test
The Lucas-Lehmer test
An improved n + 1 test, and a combined n[superscript 2] - 1 test
Divisors in residue classes
The finite field primality test
Gauss and Jacobi sums
Gauss sums test
Jacobi sums test
The primality test of Agrawal, Kayal, and Saxena (AKS test)
Primality testing with roots of unity
The complexity of Algorithm 4.5.1
Primality testing with Gaussian periods
A quartic time primality test
Exercises
Research problems
Exponential Factoring Algorithms
Squares
Fermat method
Lehman method
Factor sieves
Monte Carlo methods
Pollard rho method for factoring
Pollard rho method for discrete logarithms
Pollard lambda method for discrete logarithms
Baby-steps, giant-steps
Pollard p - 1 method
Polynomial evaluation method
Binary quadratic forms
Quadratic form fundamentals
Factoring with quadratic form representations
Composition and the class group
Ambiguous forms and factorization
Exercises
Research problems
Subexponential Factoring Algorithms
The quadratic sieve factorization method
Basic QS
Basic QS: A summary
Fast matrix methods
Large prime variations
Multiple polynomials
Self initialization
Zhang's special quadratic sieve
Number field sieve
Basic NFS: Strategy
Basic NFS: Exponent vectors
Basic NFS: Complexity
Basic NFS: Obstructions
Basic NFS: Square roots
Basic NFS: Summary algorithm
NFS: Further considerations
Rigorous factoring
Index-calculus method for discrete logarithms
Discrete logarithms in prime finite fields
Discrete logarithms via smooth polynomials and smooth algebraic integers
Exercises
Research problems
Elliptic Curve Arithmetic
Elliptic curve fundamentals
Elliptic arithmetic
The theorems of Hasse, Deuring, and Lenstra
Elliptic curve method
Basic ECM algorithm
Optimization of ECM
Counting points on elliptic curves
Shanks-Mestre method
Schoof method
Atkin-Morain method
Elliptic curve primality proving (ECPP)
Goldwasser-Kilian primality test
Atkin-Morain primality test
Fast primality-proving via ellpitic curves (fastECPP)
Exercises
Research problems
The Ubiquity of Prime Numbers
Cryptography
Diffie-Hellman key exchange
RSA cryptosystem
Elliptic curve cryptosystems (ECCs)
Coin-flip protocol
Random-number generation
Modular methods
Quasi-Monte Carlo (qMC) methods
Discrepancy theory
Specific qMC sequences
Primes on Wall Street?
Diophantine analysis
Quantum computation
Intuition on quantum Turing machines (QTMs)
The Shor quantum algorithm for factoring
Curious, anecdotal, and interdisciplinary references to primes
Exercises
Research problems
Fast Algorithms for Large-Integer Arithmetic
Tour of "grammar-school" methods
Multiplication
Squaring
Div and mod
Enhancements to modular arithmetic
Montgomery method
Newton methods
Moduli of special form
Exponentiation
Basic binary ladders
Enhancements to ladders
Enhancements for gcd and inverse
Binary gcd algorithms
Special inversion algorithms
Recursive-gcd schemes for very large operands
Large-integer multiplication
Karatsuba and Toom-Cook methods
Fourier transform algorithms
Convolution theory
Discrete weighted transform (DWT) methods
Number-theoretical transform methods
Schonhage method
Nussbaumer method
Complexity of multiplication algorithms
Application to the Chinese remainder theorem
Polynomial arithmetic
Polynomial multiplication
Fast polynomial inversion and remaindering
Polynomial evaluation
Exercises
Research problems
Book Pseudocode
References

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