The encryption in Paillier cryptosystem is like this according to Wikipedia:

  1. Let $m$ be a message to be encrypted where $m \in \mathbb{Z}_n$
  2. Select random $r$ where $r \in \mathbb{Z}_n^*$
  3. Compute ciphertext as: $c = g^m \cdot r^n \bmod n^2 $

So just calculating the inverse of ciphertext $c$ (i.e enc($m$)) will not work, because enc($m$)*enc($-m$) is equal to $r^{2n} \bmod n^2$, not $1 \bmod n^2$.

  • $\begingroup$ I tried to convert your question to proper formatting. Please edit if I mangled something. $\endgroup$
    – otus
    Oct 25 '15 at 20:17

Actually, computing the inverse modulo $n^2$ using (say) the Extended Euclidean method is exactly what you do. If $c = g^m r^n \bmod n^2$ is an encrypted version of $m$, then $c^{-1} = (g^m r^n)^{-1} = g^{-m} (r^{-1})^n$ is a representation of $-m$ (because if $r \in \mathbb{Z}^*_n$, so is $r^{-1}$)

That $enc(m) * enc(-m)$ isn't precisely 1 isn't relevant; however if we use different randomness $r$ and $s$ during the separate encryptions (as we're supposed to), it'll actually be $r^ns^n = (rs)^n$. However, there is lots of encrypted representations of 0; while 1 is such a valid representation, $(rs)^n$ is as well.


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