Prove the correctness of decryption process of Paillier cipher
First, we simplify $\mu$.
\begin{align*}
    g &amp;= (1+n)^{\alpha}\beta^{n} \bmod n^{2} \\
    g^{\lambda} &amp;= (1+n)^{\alpha\lambda}\beta^{n\lambda}\bmod n \\
    &amp;= (1+n)^{\alpha\lambda}\bmod n^{2} \\
    &amp;= (1+n\alpha\lambda)\bmod n^{2} \\
    L(g^{\lambda}\bmod n^{2}) &amp;= (\alpha\lambda)\bmod n^{2}
\end{align*}
Then, let's take a look at $L(c^{\lambda}\bmod n^{2})$.
\begin{align*}
    c &amp;= g^{m}r^{n}\bmod n^{2} \\
    c^{\lambda} &amp;= g^{m\lambda}r^{n\lambda}\bmod n^{2} \\
    &amp;= g^{m\lambda}\bmod n^{2} \\
    &amp;= (1+n\alpha\lambda)^{m}\bmod n^{2} \\
    &amp;= (1+mn\alpha\lambda)\bmod n^{2} \\
    L(c^{\lambda}\bmod n^{2}) &amp;= (m\alpha\lambda)\bmod n^{2}
\end{align*}
Hence, $\frac{L(c^{\lambda}\bmod n^{2})}{L(g^{\lambda}\bmod n^{2})}=m\bmod n$, the decryption process of Paillier cipher is correct.