Polynomial values in small subgroups of finite fields

Abstract

For a large prime $p$, and a polynomial $f$ over a finite field ${\mathbb{F}}_p$ of $p$ elements, we obtain a lower bound on the size of the multiplicative subgroup of ${\mathbb{F}}_p^{\ast }$ containing $H\ge 1$ consecutive values $f(x),x=u+1,\dots ,u+H$, uniformly over $f\in {\mathbb{F}}_p[X]$ and an $u\in {\mathbb{F}}_p$.