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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 in mathematics. Inrecent years we have witnessed a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. The purpose of this book is to introduce the reader to some of these recent developments. It should be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics. We shall focus our attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, we have not attempted 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. To limit the size of the volume we have been forced to omit some important applications of finite fields. Some of these missing applications are briefly mentioned in the Appendix along with some key references.

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 in mathematics. Inrecent years we have witnessed a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. The purpose of this book is to introduce the reader to some of these recent developments. It should be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics. We shall focus our attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, we have not attempted 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. To limit the size of the volume we have been forced to omit some important applications of finite fields. Some of these missing applications are briefly mentioned in the Appendix along with some key references.

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CONTENTS
Preface = ⅸ
Acknowledgements = xi
1 Introduction to Finite Fields and Bases = 1
  1.1 Introdution = 1
  1.2 Bases = 3
  1.3 The Enumeration of Bases = 7
  1.4 Application = 12
  1.5 References = 15
2 Factoring Polynomials over Finite Fields = 17
  2.1 Introduction = 17
  2.2 A Few Basics = 20
  2.3 Root Finding = 22
  2.4 Factoring = 26
  2.5 Factoring Multivariate Polynomials = 34
  2.6 References = 37
3 Construction of Irreducible Polynomials = 39
  3.1 Introduction = 39
  3.2 Specific Irreducible Polynomials = 40
  3.3 Irreducibility of Compositions of Polynomials = 43
  3.4 Recursive Constructions = 49
  3.5 Composed Product of Irreducible Polynomials = 55
  3.6 A General Approach = 60
  3.7 References = 65
4 Normal Bases = 69
  4.1 Introduction = 69
  4.2 Some Properties of Normal bases = 70
  4.3 Distribution of Normal Elements = 76
  4.4 Characterization of N-Polynomials = 83
  4.5 Construction of Normal Bases = 86
  4.6 Comments = 90
  4.7 References = 90
5 Optimal Normal Bases = 93
  5.1 Introduction = 93
  5.2 Constructions = 96
  5.3 Determination of all Optimal Normal Bases = 104
  5.4 An Open Problem = 110
  5.5 References = 112
6 The Discrete Logarithm Problem = 115
  6.1 Introduction = 115
  6.2 Applications = 118
  6.3 The Discrete Logarithm Problem : General Remarks = 122
  6.4 Square Root Methods = 123
  6.5 The Pohlig-Hellman Method = 124
  6.6 The Index Calculus Method = 126
  6.7 Best Algorithms = 130
  6.8 Computational Results = 130
  6.9 Discrete Logarithms and Factoring = 131
  6.10 References = 136
7 Elliptic Curves over Finite Fields = 139
  7.1 Definitions = 139
  7.2 Group Law = 141
  7.3 The Discriminant and j - Invariant = 142
  7.4 Curves over K, char(K) ≠ 2, 3 = 143
  7.5 Curves over K, char(K) ＝ 2 = 144
  7.6 Group Structure = 146
  7.7 Supersingular Curves = 148
  7.8 References = 149
8 Elliptic Curve Cryptosystems = 151
  8.1 Introduction = 151
  8.2 Singular Elliptic Curves = 152
  8.3 The Elliptic Curve Logarithm Problem = 154
  8.4 Implementation = 163
  8.5 References = 170
9 Introduction to Algebraic Geometry = 173
  9.1 Affine Varieties = 173
  9.2 Plane Curves = 175
  9.3 Projective Varieties = 177
  9.4 Projective Plane Curves = 179
  9.5 Dimension of X = 180
  9.6 Divisors on X = 181
  9.7 Differentials on X = 184
  9.8 Algebraic Curves over a Finite Field = 189
  9.9 References = 190
10 Codes From Algebraic Geometry = 191
  10.1 Introduction to Coding Theory = 192
  10.2 Algebraic Geometric Codes = 193
  10.3 Hermitian Codes = 197
  10.4 Codes From Elliptic Curves = 200
  10.5 Codes From Elliptic Corves over F2 m = 201
  10.6 Decoding Algebraic Geometric Codes = 203
  10.7 Problems = 207
  10.8 References = 208
Appendix-Other Applications = 211