Applications of finite fields /
"The theory of finite fields, whose origins can be traced back to the works of Gauss and Galois, has played a part in various branches of mathematics, in recent years there has been a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and...
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| Other Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Boston :
Kluwer Academic Publishers,
©1993.
|
| Series: | Kluwer international series in engineering and computer science ;
SECS 199. Kluwer international series in engineering and computer science. Communications and information theory. |
| Subjects: |
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| 245 | 0 | 0 | |a Applications of finite fields / |c by Alfred J. Menezes, editor ; Ian F. Blake [and others]. |
| 260 | |a Boston : |b Kluwer Academic Publishers, |c ©1993. | ||
| 300 | |a xi, 218 pages : |b illustrations ; |c 24 cm | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a unmediated |b n |2 rdamedia | ||
| 338 | |a volume |b nc |2 rdacarrier | ||
| 490 | 1 | |a The Kluwer international series in engineering and computer science ; |v SECS199. |a Communications and information theory | |
| 504 | |a Includes bibliographical references and index. | ||
| 505 | 0 | 0 | |t Preface -- |t Acknowledgements -- |g 1 |t Introduction to Finite Fields and Bases |g (starting p. 1) -- |g 1.2 |t Bases -- |g 1.3 |t The Enumeration of Bases -- |g 1.4 |t Applications -- |g 2 |t Factoring Polynomials over Finite Fields |g (starting p. 17) -- |g 2.2 |t A Few Basics -- |g 2.3 |t Root Finding -- |g 2.4 |t Factoring -- |g 2.5 |t Factoring Multivariate Polynomials -- |g 3 |t Construction of Irreducible Polynomials |g (starting p. 39) -- |g 3.2 |t Specific Irreducible Polynomials -- |g 3.3 |t Irreducibility of Compositions of Polynomials -- |g 3.4 |t Recursive Constructions -- |g 3.5 |t Composed Product of Irreducible Polynomials -- |g 3.6 |t A General Approach -- |g 4 |t Normal Bases |g (starting p. 69) -- |g 4.2 |t Some Properties of Normal bases -- |g 4.3 |t Distribution of Normal Elements -- |g 4.4 |t Characterization of N-Polynomials -- |g 4.5 |t Construction of Normal Bases -- |g 5 |t Optimal Normal Bases |g (starting p. 93) -- |g 5.2 |t Constructions -- |g 5.3 |t Determination of all Optimal Normal Bases -- |g 5.4 |t An Open Problem -- |g 6 |t The Discrete Logarithm Problem |g (starting p. 115) -- |g 6.2 |t Applications -- |g 6.3 |t The Discrete Logarithm Problem: General Remarks -- |g 6.4 |t Square Root Methods -- |g 6.5 |t The Pohlig-Hellman Method -- |g 6.6 |t The Index Calculus Method -- |g 6.7 |t Best Algorithms -- |g 6.8 |t Computational Results -- |g 6.9 |t Discrete Logarithms and Factoring -- |g 7 |t Elliptic Curves over Finite Fields |g (starting p. 139) -- |g 7.2 |t Group Law -- |g 7.3 |t The Discriminant and j-Invariant -- |g 7.4 |t Curves over K, char(K) [actual symbol not reproducible] 2, 3 -- |g 7.5 |t Curves over K, char(K) = 2 -- |g 7.6 |t Group Structure -- |g 7.7 |t Supersingular Curves -- |g 8 |t Elliptic Curve Cryptosystems |g (starting p. 151) -- |g 8.2 |t Singular Elliptic Curves -- |g 8.3 |t The Elliptic Curve Logarithm Problem -- |g 8.4 |t Implementation -- |g 9 |t Introduction to Algebraic Geometry |g (starting p. 173) -- |g 9.1 |t Affine Varieties -- |g 9.2 |t Plane Curves -- |g 9.3 |t Projective Varieties -- |g 9.4 |t Projective Plane Curves -- |g 9.5 |t Dimension of X -- |g 9.6 |t Divisors on X -- |g 9.7 |t Differentials on X -- |g 9.8 |t Algebraic Curves over a Finite Field -- |g 10 |t Codes From Algebraic Geometry |g (starting p. 191) -- |g 10.3 |t Hermitian Codes -- |g 10.4 |t Codes From Elliptic Curves -- |g 10.5 |t Codes From Elliptic Curves over F[subscript 2m] -- |g 10.6 |t Decoding Algebraic Geometric Codes -- |t Appendix -- Other Applications |g (starting p. 211) -- |t Index |g (starting p. 215) |
| 520 | |a "The theory of finite fields, whose origins can be traced back to the works of Gauss and Galois, has played a part in various branches of mathematics, in recent years there has been a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. Applications of Finite Fields introduces some of these recent developments. This book focuses attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, Applications of Finite Fields does not attempt to be encyclopedic. Among the topics studied are different methods of representing the elements of a finite field (including normal bases and optimal normal bases), algorithms for factoring polynomials over finite fields, methods for constructing irreducible polynomials, the discrete logarithm problem and its implications to cryptography, the use of elliptic curves in constructing public key cryptosystems, and the uses of algebraic geometry in constructing good error-correcting codes. This book is developed from a seminar held at the University of Waterloo. The purpose of the seminar was to bridge the knowledge of the participants whose expertise and interests ranged from the purely theoretical to the applied. As a result, this book will be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics"--Publisher description. | ||
| 650 | 0 | |a Engineering mathematics. |0 http://id.loc.gov/authorities/subjects/sh85043235 | |
| 650 | 0 | |a Finite fields (Algebra) |0 http://id.loc.gov/authorities/subjects/sh85048351 | |
| 650 | 6 | |a Corps finis. | |
| 650 | 6 | |a Mathématiques de l'ingénieur. | |
| 650 | 7 | |a Engineering mathematics. |2 fast |0 (OCoLC)fst00910601 | |
| 650 | 7 | |a Finite fields (Algebra) |2 fast |0 (OCoLC)fst00924905 | |
| 650 | 1 | 7 | |a Veldentheorie. |2 gtt |
| 650 | 7 | |a Finite fields (Algebra) |2 nli | |
| 650 | 7 | |a Error-correcting codes (Information theory) |2 nli | |
| 650 | 7 | |a Cryptography. |2 nli | |
| 650 | 7 | |a Engineering mathematics. |2 nli | |
| 650 | 7 | |a Mathématiques de l'ingénieur. |2 ram | |
| 650 | 7 | |a Corps finis. |2 ram | |
| 653 | 0 | |a Information |a Statistical mechanics | |
| 700 | 1 | |a Menezes, A. J. |q (Alfred J.), |d 1965- |0 http://id.loc.gov/authorities/names/n91082082 | |
| 700 | 1 | |a Blake, Ian F. | |
| 830 | 0 | |a Kluwer international series in engineering and computer science ; |v SECS 199. | |
| 830 | 0 | |a Kluwer international series in engineering and computer science. |p Communications and information theory. |0 http://id.loc.gov/authorities/names/n86712327 | |
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