Linear Recurring Sequences Over Finite Fields

Abstract

This thesis deals with the problem of how the elements from a finite field F of characteristic p are distributed among the various linear recurrent sequences with a given fixed characteristic polynomial fe F[x]. The first main result is a of extending the so-called classical method for solving linear recurrences in terms of the roots of f. The main difficulty is that f might have a root θ which occurs with multiplicity exceeding p-1; this is overcome by replacing the solutions θt, tθt, t2θt, ..., by the solutions θt, (t1)θt, (t2)θt, .... The other main result deals with the number N of times a given element a e F appears in a period of the sequence, and for a≠0, the result is of the form N≡0 (mod pe where e is an integer which depends upon f, but not upon the particular sequence in question. Several applications of the main results are given.