This is a part of homework assignment, and I am stuck. The RSA signature is being calculated using Chinese Remainder theorem technique. Find the detailed description here.
Public and private keys are $(N,e)$ and $(N,p,q,d)$ respectively. $N = pq$. $p, q$ being prime. $e\cdot q = 1 \bmod \Phi(N)$
- $S_p = M^d \mod p = M^{d \mod (pā1)} \bmod p$
- $S_q = M^d \mod q = M^{d \mod (qā1)} \bmod q$
- $\rm{Sign}_{(d,p,q)} (M ) = (S_{p} \cdot \beta \cdot q + S_{q} \cdot \alpha \cdot p) \bmod N$
$M$ is the message.
The problem states that if the value of $S_p$ is calculated correctly but value of $S_q$ is not. Then the receiver finds out the secret key.. We have to give the description of this receiver that figures out the secret key.
Any help will be appreciated. thanks in advance