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Question about homomorphic property of Paillier cryptosystem breaks after modulus n are taken

cn flag

Hi I am watching this slide about threshold signature: http://cyber.biu.ac.il/wp-content/uploads/2021/11/Threshold_Sinature_Schemes_Rosario_Gennaro.pdf
And in page 39 it said that the homomorphic property of Paillier is based on the message is small, that if the mod n is taken, the homomorphic property breaks.
I am trying to understand this, could anyone provide some help? I would truly appreciate it.

Hilder Vitor Lima Pereira avatar
us flag
Paillier message space is $\mathbb{Z}_n$, so the homomorphic addition produces an encryption of $m_1 + m_2 \bmod n$. *My guess* is that protocol the slides are referring to need to decrypt $m_1 + m_2$. So you need $m_1 + m_2 < n$ to guarantee that not $(m_1 + m_2 \bmod n) = m_1 + m_2$ and the decrypted value will the expected one.
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