C C ] 1 M ar 2 01 8 IDENTITY TESTING AND INTERPOLATION FROM HIGH POWERS OF POLYNOMIALS OF LARGE DEGREE OVER FINITE FIELDS

We consider the problem of identity testing and recovering (that is, interpolating) of a “hidden” monic polynomials f , given an oracle access to f(x) for x ∈ Fq , where Fq is the finite field of q elements and an extension fields access is not permitted. The naive interpolation algorithm needs de + 1 queries, where d = max{deg f, deg g} and thus requires de < q . For a prime q = p , we design an algorithm that is asymptotically better in certain cases, especially when d is large. The algorithm is based on a result of independent interest in spirit of additive combinatorics. It gives an upper bound on the number of values of a rational function of large degree, evaluated on a short sequence of consecutive integers, that belong to a small subgroup of F∗ p .