# Paillier scheme and noise growth

Does the problem of noise growth exist in the Paillier homomorphic scheme ?

• Yes. It must sometime be taken into account that the homomorphic property of the Paillier scheme is modulo $N$.
• @witdev: The value $g^x r^n \bmod n^2$ is a valid encryption of the plaintext $x$ (for any $0 < r < n$, $r$ r.p. to $n$. The multiplication of the two ciphertexts $g^x r^n \bmod n^2$ and $g^y r'^n \bmod n^2$ is thus $g^{x+y} (rr')^n \bmod n^2$, and thus is always a valid encryption of $x+y \bmod n$. This always holds, and hence there is no 'noise growth'; the interaction of the random values $r, r'$ never interferes with the decryption process (and so you don't run into the same problems you do with Lattice based systems) May 20 '20 at 16:39