Let $E$ be an elliptic curve defined over a finite field $F_q$ with prime order $n$ and $P,Q \in E$ and $k$ be private key such that $kP=Q$. Since $n$ is prime, $E$ is isomorphic to $Z_n$. Suppose $\psi$ be a map that $$\psi:E \to Z_n$$
Also, let $a,b\in Z_n$ such that $\psi(P)=a,\psi(Q)=b$. Then we have:
All recommended NIST elliptic curves(P-192,...,P-521,K-163,...) have prime orders.
Also When $\#E(F_p)=p$ we have polynomial time for computing such map. This is well known attack that is called anomalous curve attack.
What is the reason for confidence to them? Are computations speed enough?