  We consider a natural generalization of an abelian Hidden Subgroup Problem
where the subgroups and their cosets correspond to graphs of linear functions
over a finite field F with d elements. The hidden functions of the generalized
problem are not restricted to be linear but can also be m-variate polynomial
functions of total degree n>=2.
  The problem of identifying hidden m-variate polynomials of degree less or
equal to n for fixed n and m is hard on a classical computer since
Omega(sqrt{d}) black-box queries are required to guarantee a constant success
probability. In contrast, we present a quantum algorithm that correctly
identifies such hidden polynomials for all but a finite number of values of d
with constant probability and that has a running time that is only
polylogarithmic in d.
