Let $A \subset \mathbb{Z}/p$, let $f$ be a function on $\mathbb{Z}/p$ and let $B:=\{f(a): a \in A\}$.

Can we conclude that $|A+B|$ is large if $f$ is a sufficiently "nice" function? For instance say that $f(a)=a^2$. Then can we say that if $|A|=1000\sqrt{p}$ or even if $|A|=\frac{p}{\log^{*}(p)}$, then $|A+B| \ge p/100$?

The idea should be that $A+B$ is only small if $A,B$ are related by some additive structure, and if $f$ is a sufficiently random function then it should kill any structure so that $A+B$ should be large.