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In standard Paillier encryption Property 1 really is: $m_1=D(c_1)\text{ and }m_2=D(c_2)\implies D(c_1\cdot c_2\bmod n^2) = m_1+m_2\bmod n$. Property 2 does not hold (but see final off-topic note). As a consequence of property 1, for all $k$ in $\mathbb Z$, it holds $D({c_1}^k\bmod n^2)\ =\ m_1\cdot k\bmod n$. Proof for positive $k$ can be by induction, ...

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