Distribution of inverses in polynomial rings

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