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In the original paper of Paillier, lemma 1 shows why $n$ must divide the order of $g$. What I don't understand in the proof of this lemma is why $g^{x_2-x_1}(y_2/y_1)^n$ implies $g^{\lambda(x_2-x_1)}$. Where does this result come from?

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This is obtained by raising to $\lambda=\lambda(n)$: since the order of any element in $\mathbb{Z}_{n^2}$ divides $n\cdot\lambda$, the second part cancels out:

$$\begin{align} g^{x_1-x_2}\cdot(y_2/y_1)^n=1\bmod{n^2} &\Leftrightarrow\\ g^{(x_1-x_2)\cdot\lambda}\cdot(y_2/y_1)^{n\cdot\lambda}=1\bmod{n^2}&\Leftrightarrow\\ g^{(x_1-x_2)\cdot\lambda}\cdot 1=1\bmod{n^2}&\Leftrightarrow\\ g^{(x_1-x_2)\cdot\lambda}=1\bmod{n^2} \end{align}$$

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  • $\begingroup$ Thx! I didn't see that. $\endgroup$ – mip Jun 14 at 16:13

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