Quadratic distances in even dimensions over prime fields

Let $p$ be an odd prime, let $m\ge 1$ be an integer, and let $Q$ be a nondegenerate quadratic form on ${\mathbb{F}}_p^{2m}$ with Witt index $m-1$. For a nonempty set $E\subseteq {\mathbb{F}}_p^{2m}$, write ${\mathrm{\Delta }}_Q(E)=\{Q(x-y):x,y\in E\}$. We prove that, whenever $\mathrm{\mid }E\mathrm{\mid }\ge p^m$, $$\mathrm{\mid }{\mathrm{\Delta }}_Q(E)\mathrm{\mid }\gg \frac{p}{\log (2+p^{m+1}\mathrm{/}\mathrm{\mid }E\mathrm{\mid })},$$ with an absolute implied constant independent of $p$, $m$ and $Q$. In the planar case $m=1$, we also prove that $$\mathrm{\mid }{\mathrm{\Delta }}_Q(E)\mathrm{\mid }\gg \frac{\mathrm{\mid }E\mathrm{\mid }}{\log (2\mathrm{\mid }E\mathrm{\mid })},(1\le \mathrm{\mid }E\mathrm{\mid }\le p),$$ which is optimal up to a logarithmic factor.