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Let $p$ and $q$ are Sophie-Germain primes such that $p=2p'+1$ and $q=2q'+1$. Also let $n=pq$ and $n'=p'q'$. In Section 8.2.1 of this paper, the internal direct product of $\mathbb{Z}_{n^2}^*$ is shown as $$\mathbb{G}_{n}\cdot\mathbb{G}_{n'}\cdot\mathbb{G}_{2}\cdot T$$ where $\mathbb{G}_{\tau}$ is the cyclic group with the order $\tau$ and $T$ is the subgroup generated by $-1\text{ mod }n^2$. Furthermore, the paper says that this decomposition is unique except $\mathbb{G}_{2}$ where there are two possible choices. However, as far as I know, there is a unique cyclic group with order 2. Hence, I think that $\mathbb{G}_{2}$ must also be unique. What am I missing there?

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Let $g$ be such that $g\equiv 1\pmod {p^2}$ and $g\equiv -1\pmod {q^2}$, there is a unique solution to this by the Chinese remainder theorem (and this solution is not 0, 1 or -1). We see that $\langle g\rangle$ is a cyclic group of order 2, because $g^2\equiv 1\pmod {p^2}$ and $g^2\equiv 1\pmod {q^2}$ which implies that $g^2\equiv 1\pmod {n^2}$.

Likewise let $h$ be such that $h\equiv -1\pmod {p^2}$ and $h\equiv 1\pmod {q^2}$, there is a unique solution to this by the Chinese remainder theorem. We see that $\langle h\rangle$ is also a cyclic group of order 2, but the groups are distinct.

Note that $g=-h$ and vice-versa.

The group $\mathbb G_2$ can be taken as either $\langle g\rangle$ or $\langle h\rangle$.

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  • $\begingroup$ Let me ask a further question which is both related to the paper and the previous question. Let $g'\leftarrow \mathbb{Z}_{n^2}^*$ and $g=(g')^{2n}$. Then, considering our direct product, $g$ must be a member of $\mathbb{G}_{n'}$, right? $\endgroup$
    – kentakenta
    Feb 13, 2022 at 14:13
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    $\begingroup$ Yes (assuming that you've not picked a pathological case such as $p'=q$). $\endgroup$
    – Daniel S
    Feb 13, 2022 at 14:34

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