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Product Code:  MMONO/232 
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Electronic ISBN:  9781470418144 
Product Code:  MMONO/232.E 
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Book DetailsTranslations of Mathematical MonographsVolume: 232; 2007; 243 ppMSC: Primary 11; Secondary 94;
Algorithmic number theory is a rapidly developing branch of number theory, which, in addition to its mathematical importance, has substantial applications in computer science and cryptography. Among the algorithms used in cryptography, the following are especially important:
 algorithms for primality testing;
 factorization algorithms for integers and for polynomials in one variable;
 applications of the theory of elliptic curves;
 algorithms for computation of discrete logarithms;
 algorithms for solving linear equations over finite fields;
 algorithms for performing arithmetic operations on large integers.
The book describes the current state of these and some other algorithms. It also contains extensive bibliography. For this English translation, additional references were prepared and commented on by the author.ReadershipGraduate students and research mathematicians interested in algorithmic number theory and its applications.

Table of Contents

Chapters

Primality testing and construction of large primes

Factorization of integers with exponential complexity

Factorization of integers with subexponential complexity

Application of elliptic curves to primality testing and factorization of integers

Algorithms for computing discrete logarithm

Factorization of polynomials over finite fields

Reduced lattice bases and their applications

Factorization of polynomials over the field of rational numbers with polynomial complexity

Discrete Fourier transform and its applications

Highprecision integer arithmetic

Solving systems of linear equations over finite fields

Facts from number theory


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Algorithmic number theory is a rapidly developing branch of number theory, which, in addition to its mathematical importance, has substantial applications in computer science and cryptography. Among the algorithms used in cryptography, the following are especially important:
 algorithms for primality testing;
 factorization algorithms for integers and for polynomials in one variable;
 applications of the theory of elliptic curves;
 algorithms for computation of discrete logarithms;
 algorithms for solving linear equations over finite fields;
 algorithms for performing arithmetic operations on large integers.
The book describes the current state of these and some other algorithms. It also contains extensive bibliography. For this English translation, additional references were prepared and commented on by the author.
Graduate students and research mathematicians interested in algorithmic number theory and its applications.

Chapters

Primality testing and construction of large primes

Factorization of integers with exponential complexity

Factorization of integers with subexponential complexity

Application of elliptic curves to primality testing and factorization of integers

Algorithms for computing discrete logarithm

Factorization of polynomials over finite fields

Reduced lattice bases and their applications

Factorization of polynomials over the field of rational numbers with polynomial complexity

Discrete Fourier transform and its applications

Highprecision integer arithmetic

Solving systems of linear equations over finite fields

Facts from number theory