CONTENTS
Series Editor's Preface ⅸ
Preface = ⅹ
Acknowledgements = xi
Notations = ⅸ
Introduction = 1
Chapter 1. Polynomial Factorization = 7
1. Univariate factorization = 7
2. Multivariate factorization = 16
3. Other polynomial decompositions = 20
Chapter 2. Finding irreducible and primitive polyziginials = 21
1. Construction of irreducible polynomials = 21
2. Construction of primitive polynomials = 27
Chapter 3. The distribution of irreducible and priniitive polynomials = 30
1. Distribution of irreducible and primitive polynomials = 30
2. Irreducible and primitive polynomials of a given height and weight = 42
3. Sparse polynomials = 46
4. Applications to algebraic number fields = 47
Chapter 4. Bases and computation in finite fields = 49
1. Construction of some special bases for finite fields = 49
2. Discrete logarithm and Zech's logarithm = 54
3. Polynomial multiplication and multiplicative complexity infinite fields = 56
4. Other algorithms in finite fields = 64
Chapter 5. Coding theory and algebraic curves = 72
1. Codes and points on algebraic curves = 72
2. Codes and exponential sums = 86
3. Codes and lattice packings and coverings = 92
Chapter 6. Elliptic curves = 99
1. Some general properties = 99
2. Distribution of primitive points on elliptic curves = 105
Chapter 7. Recurrent sequences in finite fields and leylic lineir codes = 109
1. Distribution of values of recurrent sequences = 109
2. Applications of recurrent sequences = 113
3. Cyclic codes and recurrent sequences = 116
Chapter 8. Finite fields and discrete mathematics = 122
1. Cryptography and permutation polynomials = 122
2. Graph theory, combinatorics, Boolean functions = 129
3. Enumeration problems in finite fields = 136
Chapter 9. Congruences = 139
1. Optimal coefficients and pseudo-random numbers = 139
2. Residues of exponential functions = 143
3. Modular arithmetic = 148
4. Other applications = 150
Chapter 10. Some related problems = 153
1. Integer factorization, primality testing and the greatest common divisor = 153
2. Computational algebraic number theory = 155
3. Algebraic complexity theory = 156
4. Polynomials with integer coefficients = 159
Appendix 1 = 161
Appendix 2 = 164
Appendix 3 = 165
Addendum = 166
References = 191
Index = 238
