RSA: Fast factorization of N if d and e are known

Question

I stumbled across this paragraph in a paper:

Hence, user b cannot decrypt C directly. But using e and d , user b can quickly factor N.

How is it possible to speedup the prime factorization when knowing e (public key) and d (private key)?

For clarification:

RSA provides us with these equations:

$n = pq$

$\phi = (p-1)(q-1)$

$gcd(e, \phi) = 1$

$de = 1\pmod{\phi}$

In order to determine $p$ and $q$ an attacker has to factor n which is not feasible. However the paper stated that it is easy to reconstruct $p$ and $q$ when a person knows both (his) private and public keys.

Answer 1

It is not difficult to prove that one can factor $\rm N$ in polynomial time given any multiple of $\rm \phi(N)$ (here $\rm \:de-1\:$).$\ $ See, for example, Gary Miller: Riemann's hypothesis and tests for primality. 1976

NOTE $\ $ This fact was well-known to the discoverers of RSA. Indeed they mention it explicitly in section IX of the original 1978 paper on RSA, which is quite readable.