Elliptic curve with prime subgroup equal to field size
It seems that curves like that do not exist.

Hasse's bound states that $|N-(p+1)| \le 2\sqrt p$, where $N=\#E$. Writing the order of the group as $N = ph$, you get:

$|ph-(p+1)| \le 2\sqrt p \rightarrow$

$|p(h-1)-1)| \le 2\sqrt p \rightarrow$

$|\sqrt p(h-1)-\frac{1}{\sqrt p}| \le 2$

The only way this inequality is true for $p &gt; 5$ is if $h = 1$, i.e., $\#E=p$.