Noisy interpolation of sparse polynomials in finite fields

We consider a polynomial analogue of the hidden number problem introduced by Boneh and Venkatesan, namely the sparse polynomial noisy interpolation problem of recovering an unknown polynomial f(X) [InlineMediaObject not available: see fulltext.][X] with at most w non-zero terms from approximate values of f(t) at polynomially many points t [InlineMediaObject not available: see fulltext.] selected uniformly at random. We extend the polynomial time algorithm of the first author for polynomials f(X) of sufficiently small degree to polynomials of almost arbitrary degree. Our result is based on a combination of some number theory tools such as bounds of exponential sums and the number of solutions of congruences with the lattice reduction technique. The new idea is motivated by Waring's problem and uses a recent bound on exponential sums of Cochrane, Pinner, and Rosenhouse.