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I was asked this by my professor and I didn't understand the reasoning behind it. If Alice has a the key pair $(p_a, n_a)$ and Bob has the key pair $(p_b, n_b)$, why do $n_a$ and $n_b$ have to be relatively prime?

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In RSA, the modulus is computed as $n=pq$ where $p$ and $q$ are prime. Given two moduli, if they have no primes in common, then the GCD is $1$ and they are relatively prime. If they share a prime factor, then the GCD will reveal that prime factor. Therefore, anyone who knows the two public keys can factorize the moduli and break security.

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  • $\begingroup$ Ahh that makes total sense. So that means p and q need to be unique to everyone? Surely there will be some people who choose similar primes between themselves. I guess there is a sort of "herd protection"? $\endgroup$ – n0pe Nov 5 '14 at 12:57
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    $\begingroup$ Theoretically, if good random number generators are used, there are enough primes to ensure this won't happen. In practice, however, it has been the case that this has happened (see this). $\endgroup$ – mikeazo Nov 5 '14 at 13:42
  • $\begingroup$ An other good reference: smartfacts.cr.yp.to $\endgroup$ – ddddavidee Nov 5 '14 at 20:59

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