The Boneh-DeMillo-Lipton fault attack (RSA CRT)

Question

I am trying to understand the Boneh-DeMillo-Lipton fault attack on RSA CRT signature.

Suppose that we sign a message $m$ with RSA-CRT :

$d_p = d \bmod (p-1)$ and $d_q = d \bmod (q-1)$

$s_p = h(m)^{d_p} \bmod p$ and $s_q \not= h(m)^{d_q} \bmod q$ where $h$ is a hash function

By CRT we get $s$ the signature modulo $n=pq$

I don't understand the following assertion :

As $s^e \equiv h(m) \pmod p$ but $s^e \not \equiv h(m) \pmod q$ we can factorize $n$ by $p=GCD(s^e - h(m), n)$.

Can somebody explain to me why $p$ is the greatest common divisor between $s^e - h(m)$ and $n$?

Answer 1

$q$ does not divide $s^e-h(m)$, but $p$ does, so since the gcd must divide both $s^e-h(m)$ and $n$ it's $p$. To be even more explicit, we know that $p$ divides both $s^e-h(m)$ and $n$. The only larger divisor of $n$ that is also divisible by $p$ is $n$ itself, but if $n$ would divide $s^e-h(m)$, then $q$ would also divide $s^e-h(m)$, which we already assumed does not hold.