# Finding the random $r$ in a Paillier encrypted message with knowledge of the private key.

In the Paillier cryptosystem, suppose that I know a Ciphertext encrypted with some unknown random $r$ i.e.

$$C = (g^m r^n) \bmod n^2$$

I know $g, n$, the prime factorization of $n$, i.e., $pq$. I also know the private key $-$ hence I know the message $m$. Would there be a way for me to recover $r$ easily?

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 – mikeazo Sep 12 '12 at 14:41