Distribution of Inverses in Polynomial Rings

NOTICE: this is the author's version of a work that was accepted for publication in Indagationes Mathematicae. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Indagationes Mathematicae, Volume 12, Issue 3, (2001), Pages 303-315. doi:10.1016/S0019-3577(01)80012-4. http://www.elsevier.com/locate/indagLet IFp be the finite field with p elements, and let F(X) ∈ IFp[X] be a square-free polynomial. We show that in the ring R = IFp[X]/F(X), 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