**Translations of Mathematical Monographs**

2007;
243 pp;
Hardcover

MSC: Primary 11;
Secondary 94

Print ISBN: 978-0-8218-4090-0

Product Code: MMONO/232

List Price: $103.00

AMS Member Price: $82.40

MAA Member Price: $92.70

**Electronic ISBN: 978-1-4704-1814-4
Product Code: MMONO/232.E**

List Price: $97.00

AMS Member Price: $77.60

MAA Member Price: $87.30

#### Supplemental Materials

# Number-Theoretic Algorithms in Cryptography

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*O. N. Vasilenko*

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.

#### Readership

Graduate students and research mathematicians interested in algorithmic number theory and its applications.

#### Table of Contents

# Table of Contents

## Number-Theoretic Algorithms in Cryptography

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the English translation ix10
- Preface xi12
- Notation xiii14
- Primality testing and construction of large primes 116
- Factorization of integers with exponential complexity 3550
- Factorization of integers with subexponential complexity 5166
- Application of elliptic curves to primality testing and factorization of integers 7388
- Algorithms for computing discrete logarithm 91106
- Factorization of polynomials over finite fields 115130
- Reduced lattice bases and their applications 135150
- Factorization of polynomials over the field of rational numbers with polynomial complexity 159174
- Discrete Fourier transform and its applications 175190
- High-precision integer arithmetic 187202
- Solving systems of linear equations over finite fields 203218
- Facts from number theory 215230
- Bibliography 223238
- Index 241256
- Back Cover Back Cover1266