Introduction to Finite Fields and Their ApplicationsThe theory of finite fields is a branch of modern algebra that has come to the fore in recent years because of its diverse applications in such areas as combinatorics, coding theory, cryptology and the mathematical study of switching circuits. The first part of this book presents an introduction to this theory, emphasizing those aspects that are relevant for application. The second part is devoted to a discussion of the most important applications of finite fields, especially to information theory, algebraic coding theory, and cryptology. There is also a chapter on applications within mathematics, such as finite geometries, combinatorics and pseudo-random sequences. The book is designed as a graduate level textbook; worked examples and copious exercises that range from the routine, to those giving alternative proofs of key theorems, to extensions of material covered in the text, are provided throughout. |
Contents
Structure of Finite Fields | 44 |
Polynomials over Finite Fields | 76 |
Factorization of Polynomials | 132 |
Exponential Sums | 166 |
Linear Recurring Sequences | 189 |
Theoretical Applications of Finite Fields | 256 |
Algebraic Coding Theory | 305 |
Cryptology | 344 |
Tables | 374 |
| 399 | |
List of Symbols | 406 |
Common terms and phrases
a₁ additive character algebraic algorithm BCH code binary c₁ canonical factorization character of F characteristic polynomial code word coefficients compute cryptosystem cyclic cyclotomic polynomial d₁ defined Definition deg(f denote discrete logarithms divides elements of F Example F₁ F₂ F₂[x factor of f(x field F finite fields follows gcd(f(x Goppa code greatest common divisor homogeneous linear recurring identity irreducible factors irreducible polynomials kth-order L₁(x least period Lemma Let f linear recurrence relation linear recurring sequence matrix maximal period sequence minimal polynomial mod f(x modulo monic monic irreducible polynomials nontrivial obtain ord(f parity-check matrix polynomial f(x polynomial over F polynomials in F,[x positive degree positive integer primitive element primitive polynomial Prove q-polynomial recurrence relation relatively prime residue class ring root of f root of unity roots of f(x S₁ satisfies sequence in F splitting field subfield subgroup vector space
Popular passages
Page 404 - Q = the set of rational numbers R = the set of real numbers C = the set of complex numbers Even if we can list the elements of a set, it may not be practical to do so.