  Motivated by some algorithmic problems, we give lower bounds on the size of
the multiplicative groups containing rational function images of
low-dimensional affine subspaces of a finite field~$\mathbb{F}_{q^n}$
considered as a linear space over a subfield $\mathbb{F}_q$. We apply this to
the recently introduced algorithmic problem of identity testing of "hidden"
polynomials $f$ and $g$ over a high degree extension of a finite field, given
oracle access to $f(x)^e$ and $g(x)^e$
