Values of rational functions in small subgroups of finite fields and the identity testing problem from powers

Abstract

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 [Formula: see text] considered as a linear space over a subfield [Formula: see text]. We apply this to the recently introduced algorithmic problem of identity testing of “hidden” polynomials [Formula: see text] and [Formula: see text] over a high degree extension of a finite field, given oracle access to [Formula: see text] and [Formula: see text].