I'm reading Twenty Years of Attacks on the RSA Cryptosystem by Dan Boneh and trying to understand the proof of the Fact 1 on page 3.
Fact 1: Let $(N,e)$ be an RSA public key. Given the private key $d$, one can efficiently factor the modulus $N = pq$.
Dan says that since $\phi(N)=(p-1)(q-1)$ is even, $k=de-1=2^t r$ with $r$ odd and $t\geq1$. $k$ is a multiple of $\phi(N)$. Well, I tried to derive that formula for $k$, but didn't succeed. Any ideas?