## 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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

a e F additive character algebraic algorithm BCH code binary calculate called canonical factorization character of F characteristic polynomial code word coefficients compute coset cryptosystem cyclotomic polynomial decoding defined Definition denote discrete logarithms divides elements of F Example extension field field F finite fields follows given Goppa code greatest common divisor homogeneous linear recurring identity impulse response sequence initial state vector irreducible factors irreducible over F irreducible polynomials kth-order least period Lemma Let F linear recurrence relation linear recurring sequence maximal period sequence minimal polynomial modulo monic monic irreducible polynomials multiplicative character nomial nontrivial nonzero polynomial obtain parity-check matrix polynomial f(x polynomial in fq[x polynomial over F positive degree primitive element primitive polynomial Prove recurrence relation relatively prime residue class ring root of unity 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.