Introduction to Finite Fields and Their Applications
The 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.
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according additive algebraic algorithm applications basis binary calculate called canonical Chapter character characteristic polynomial coefficients column common compute condition consider constant contains corresponding cryptosystem cyclic defined Definition degree denote determined discrete distinct divides divisible divisor elements of F equal equation equivalent error Example exists extension factors field F finite fields follows function given hence homogeneous linear recurring ideal identity implies integer least period Lemma linear recurrence relation linear recurring sequence matrix method minimal polynomial modulo multiplicative nonzero normal obtain operations period sequence plane points polynomial over F positive integer possible prime primitive element Proof properties Prove q-polynomial relatively residue respect result ring roots satisfies sequence in F shows subgroup Suppose symbolically Theorem theory uniquely values vector space write yields