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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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Breaking the Generalized Diffie-Hellman (GDH) assumption is known to imply Factoring for Blum-integers. This is a result by Biham et al. Since the (non)-equivalence of the RSA and Factoring assumptions is one of the biggest open questions in the RSA literature, it would be really surprising if there was a reduction from GDH to the RSA assumption, because ...


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