Distribution of inverses in polynomial rings

Let F-p be the finite field with p elements, and let F (X) is an element of F-p[X] be a square-free polynomial. We show that in the ring R = F-p[X]/F(X), the inverses of polynomials of small height are uniformly distributed. We also show that for any set L subset of R of very small cardinality, for almost all G is an element of R the set of inverses {(G + f) (1)\f is an element of L} are uniformly distributed. These questions are motivated by applications to the NTRU cryptosystem.