On Consecutive Primitive Roots And Large Families Of Pseudorandom Subsets Constructed By Using Primitive Roots

Let p>2 be a prime,and let Gp be the set of the primitive roots modulo p.Numerous papers have been written on the distribution of the consecutive primitive roots modulo p,whose results point to the direction that the subset Gp is of random type.Moreover,The need for pseudorandom subsets arises in many cryptographic applications,so it is vital to construct new subsets con-stantly to satisfy the demand from different areas.In this paper,we further studied the distribution of the consecutive primitive roots and the pseudoran-domness of primitive roots modulo p.Moreover,we construct Multi-dimensional pseudorandom subsets arising from Golomb's conjectures.Main results are as follows:1.Let p be a prime,f(x)?p[x]with degree D>1.Let k? 2 be an integer,and let l1,12,...,lk be distinct elements in Fp.Suppose that at least one of the following conditions holds:(i)f(x)is irreducible,(ii)f(x)has no multiple zero in (?)p,D<p and k=2,(iii)f(x)has no multiple zero in (?)p and(4k)D<p.We prove that for all primes p>max {e23k,(kD)27} there exists n ? Fp such that all are primitive roots modulo p.2.We answer a conjecture of Dartyge,Sarkozy and Szalay,and study the pseudorandom properties of some subsets and multi-dimensional subsets con-structed by using primitive roots in finite fields.3.we construct multi-dimensional pseudorandom subsets arising from Golomb's conjectures.Moreover,for these subsets7we prove that it's pseudoran-domness are not "good" when taken out the cocdition of Golomb's conjectures.