Identity Testing from High Powers of Polynomials of Large Degree over Finite Fields.

We consider the problem of identity testing of two hidden monic polynomials $f$ and $g$, given an oracle access to $f(x)^e$ and $g(x)^e$ for $xin{mathbb F}_q$, where ${mathbb F}_q$ is the finite field of $q$ elements (an extension fields access is not permitted). naive interpolation algorithm needs $de+1$ queries, where $d =max{mathrm{deg}, f, mathrm{deg}, g}$ and thus requires $ deu003cq$. 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 ${mathbb F}_p^*$.