عنوان

Computational and Algorithmic Problems in Finite Fields

(Number (ISBN

940111806X

(Number (ISBN

9789401118064

Number

b594926

Title Proper

Computational and Algorithmic Problems in Finite Fields

General Material Designation

[Book]

First Statement of Responsibility

by Igor E. Shparlinski.

Place of Publication, Distribution, etc.

Dordrecht

Name of Publisher, Distributor, etc.

Springer Netherlands

Date of Publication, Distribution, etc.

1992

Specific Material Designation and Extent of Item

(xii, 240 pages)

Series Title

Mathematics and Its Applications (Soviet Series), 88.

Text of Note

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.

Text of Note

This volume presents an exhaustive treatment of computation and algorithms for finite fields. Topics covered include polynomial factorization, finding irreducible and primitive polynomials, distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types, and new applications of finite fields to other araes of mathematics. For completeness, also included are two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number generators, modular arithmetic etc.), and computational number theory (primality testing, factoring integers, computing in algebraic number theory, etc.) The problems considered here have many applications in computer science, coding theory, cryptography, number theory and discrete mathematics. The level of discussion presuppose only a knowledge of the basic facts on finite fields, and the book can be recommended as supplementary graduate text. For researchers and students interested in computational and algorithmic problems in finite fields.

Algebra -- Data processing.

Information theory.

Mathematics.

Class number

Book number

by Igor E. Shparlinski.

Igor E Shparlinski

Electronic name

[Book]

Y