RSA - Finding all possible encryption exponents Consider the standard RSA-algorithm: We have chosen two primes $p$ and $q $ such that $n:=p \cdot q$ and computed $\varphi(n)$. We now need to choose the public key $e$ such that 

$$1&lt;e&lt;\varphi(n) \text{ and } gcd(e,\varphi(n)) = 1$$

If I am not mistaken there should be $\varphi(\varphi(n))-1$ many possible $e$'s ; we have to add $-1$ since we want to exclude $e \ne 1$. 

Is there a "clever" algorithm to find all possible $e$'s other than to apply Euclid's algorithm on all numbers below $\varphi(n)$ and checks whether they are coprime to it?
We know that $\varphi(x) \ge\sqrt{\frac{x}{2}}$. The size of $\varphi(n)$ is roughly the same of $n$, so $\varphi(\varphi(n))-1$ is still a intractably large number. This means finding all possible e's is infeasible because there are too many. No matter what algorithm you use, there is no way you can find all of them in polynomial time (in the security parameter).