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Description:

Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory. This edition now includes over 150 new exercises, ranging from the routine to the challenging, that flesh out the… More material presented in the body of the text, and which further develop the theory and present new applications. The material has also been reorganized to improve clarity of exposition and presentation. Ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.Less

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Book details

Edition: 2nd Copyright year: 2008 Publisher: Cambridge University Press Publication date: 12/4/2008 Binding: Hardcover Pages: 600 Size: 6.69" wide x 9.61" long x 1.30" tall Weight: 2.684

AuthorTable of Contents

Victor Shoup is a Professor in the Department of Computer Science at the Courant Institute of Mathematical Sciences, New York University.

Preface

Preliminaries

Basic properties of the integers

Divisibility and primality

Ideals and greatest common divisors

Some consequences of unique factorization

Congruences

Equivalence relations

Definitions and basic properties of congruences

Solving linear congruences

The Chinese remainder theorem

Residue classes

Euler's phi function

Euler's theorem and Fermat's little theorem

Quadratic residues

Summations over divisors

Computing with large integers

Asymptotic notation

Machine models and complexity theory

Basic integer arithmetic

Computing in Z<sub>n</sub>

Faster integer arithmetic (*)

Notes

Euclid's algorithm

The basic Euclidean algorithm

The extended Euclidean algorithm

Computing modular inverses and Chinese remaindering

Speeding up algorithms via modular computation

An effective version of Fermat's two squares theorem

Rational reconstruction and applications

The RSA cryptosystem

Notes

The distribution of primes

Chebyshev's theorem on the density of primes

Bertrand's postulate

Mertens' theorem

The sieve of Eratosthenes

The prime number theorem ...and beyond

Notes

Abelian groups

Definitions, basic properties, and examples

Subgroups

Cosets and quotient groups

Group homomorphisms and isomorphisms

Cyclic groups

The structure of finite abelian groups (*)

Rings

Definitions, basic properties, and examples

Polynomial rings

Ideals and quotient rings

Ring homomorphisms and isomorphisms

The structure of Z<sup>*</sup><sub>n</sub>

Finite and discrete probability distributions

Basic definitions

Conditional probability and independence

Random variables

Expectation and variance

Some useful bounds

Balls and bins

Hash functions

Statistical distance

Measures of randomness and the leftover hash lemma (*)

Discrete probability distributions

Notes

Probabilistic algorithms

Basic definitions

Generating a random number from a given interval

The generate and test paradigm

Generating a random prime

Generating a random non-increasing sequence

Generating a random factored number

Some complexity theory

Notes

Probabilistic primality testing

Trial division

The Miller-Rabin test

Generating random primes using the Miller-Rabin test

Factoring and computing Euler's phi function

Notes

Finding generators and discrete logarithms in Z<sup>*</sup><sub>p</sub>

Finding a generator for Z<sup>*</sup><sub>p</sub>

Computing discrete logarithms in Z<sup>*</sup><sub>p</sub>

The Diffie-Hellman key establishment protocol

Notes

Quadratic reciprocity and computing modular square roots

The Legendre symbol

The Jacobi symbol

Computing the Jacobi symbol

Testing quadratic residuosity

Computing modular square roots

The quadratic residuosity assumption

Notes

Modules and vector spaces

Definitions, basic properties, and examples

Submodules and quotient modules

Module homomorphisms and isomorphisms

Linear independence and bases

Vector spaces and dimension

Matrices

Basic definitions and properties

Matrices and linear maps

The inverse of a matrix

Gaussian elimination

Applications of Gaussian elimination

Notes

Subexponential-time discrete logarithms and factoring

Smooth numbers

An algorithm for discrete logarithms

An algorithm for factoring integers

Practical improvements

Notes

More rings

Algebras

The field of fractions of an integral domain

Unique factorization of polynomials

Polynomial congruences

Minimal polynomials

General properties of extension fields

Formal derivatives

Formal power series and Laurent series

Unique factorization domains (*)

Notes

Polynomial arithmetic and applications

Basic arithmetic

Computing minimal polynomials in F[x]/(f)(I)

Euclid's algorithm

Computing modular inverses and Chinese remaindering

Rational function reconstruction and applications

Faster polynomial arithmetic (*)

Notes

Linearly generated sequences and applications

Basic definitions and properties

Computing minimal polynomials: a special case

Computing minimal polynomials: a more general case

Solving sparse linear systems

Computing minimal polynomials in F[X]/(f)(II)

The algebra of linear transformations (*)

Notes

Finite fields

Preliminaries

The existence of finite fields

The subfield structure and uniqueness of finite fields

Conjugates, norms and traces

Algorithms for finite fields

Tests for and constructing inrreducible polynomials

Computing minimal polynomials in F[X](f)(III)

Factoring polynomials: square-free decomposition

Factoring polynomials: the Cantor-Zassenhaus algorithm

Factoring polynomials: Berlekamp's algorithm

Deterministic factorization algorithms (*)

Notes

Deterministic primality testing

The basic idea

The algorithm and its analysis

Notes

Appendix: Some useful facts

Bibliography

Index of notation

Index

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