Distribution of Elements on Cycles of Linear Recurrences over Galois Field

Abstract

Let y be a linear recurring sequence (LRS) over the field P of q = p ^s elements, which has irreducible characteristic polynomial of degree m and period$\Delta = \frac{{{q^m} - 1}}{d}$ where d ∈ Z divides q ^m − 1 and p ^t + 1 for some t ∈ N. For this sequence v we described possible frequences of appearance of element a ∈ P in the cycle (v(0), v(1),..., v(Δ − 1)). The proofs are based on properties of Gauss sums and generalize the results of works [2, 3].