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Supposing that the (3 total) primes are kept secret? Does the reuse of $p_1$ allow an attacker to compromise $n_1$ and $n_2$ if the attacker guesses that both were generated with a shared prime between them (each having one unique prime)?

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Yes, this trivially compromises them. Simply compute the gcd of $n_1$ and $n_2$, which will return $p_1$ (assuming $q_1 \neq q_2$). The gcd can be computed efficiently using Euclid's algorithm.

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