Computational and Algorithmic Problems in Finite Fields

Inhaltsverzeichnis

• 1. Polynomial Factorization.
• 1. Univariate factorization.
• 2. Multivariate factorization.
• 3. Other polynomial decompositions.
• 2. Finding irreducible and primitive polynomials.
• 1. Construction of irreducible polynomials.
• 2. Construction of primitive polynomials.
• 3. The distribution of irreducible and primitive polynomials.
• 1. Distribution of irreducible and primitive polynomials.
• 2. Irreducible and primitive polynomials of a given height and weight.
• 3. Sparse polynomials.
• 4. Applications to algebraic number fields.
• 4. Bases and computation in finite fields.
• 1. Construction of some special bases for finite fields.
• 2. Discrete logarithm and Zech’s logarithm.
• 3. Polynomial multiplication and multiplicative complexity in finite fields.
• 4. Other algorithms in finite fields.
• 5. Coding theory and algebraic curves.
• 1. Codes and points on algebraic curves.
• 2. Codes and exponential sums.
• 3. Codes and lattice packings and coverings.
• 6. Elliptic curves.
• 1. Some general properties.
• 2. Distribution of primitive points on elliptic curves.
• 7. Recurrent sequences in finite fields and leyelic linear codes.
• 1. Distribution of values of recurrent sequences.
• 2. Applications of recurrent sequences.
• 3. Cyclic codes and recurrent sequences.
• 8. Finite fields and discrete mathematics.
• 1. Cryptography and permutation polynomials.
• 2. Graph theory, combinatorics, Boolean functions.
• 3. Enumeration problems in finite fields.
• 9. Congruences.
• 1. Optimal coefficients and pseudo-random numbers.
• 2. Residues of exponential functions.
• 3. Modular arithmetic.
• 4. Other applications.
• 10. Some related problems.
• 1. Integer factorization, primality testing and the greatest common divisor.
• 2. Computational algebraic number theory.
• 3. Algebraic complexity theory.
• 4. Polynomials with integer coefficients.
• Appendix 1.
• Appendix 2.
• Appendix 3.
• Addendum.
• References.