# Can two different pairs of RSA key have the same modulus?

Can $n=pq$ be part of two different pairs of RSA keys?

If such keys exist, say $(e_1,n)$ and $(e_2,n)$, how are they related? What will be the security concerns for the two users?

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I edited your comment to have $e_1$ and $e_2$ instead of $e_1$ twice. If that was not your intention, feel free to fix. –  mikeazo May 2 '12 at 11:40
thanks @mikeazo ...I meant $e_1$ and $e_2$ ... –  AIB May 2 '12 at 18:22
Related to your question is whether then moduli of different public keys should be coprime (they should!!), but in real life this is sometimes not the case, see freedom-to-tinker.com/blog/nadiah/… –  j.p. May 11 '12 at 16:06

The users will be able to read each other's messages (even though they can have different private keys, say $d_1$ and $d_2$). This is because knowledge of $d_i$ is sufficient to factor $N$, thus allowing that party to compute the other party's private key. This was detailed by Boneh in his analysis of RSA attacks.

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Two properties of RSA are important here:

• If you know $p$ and $q$, you can reverse RSA encryption for arbitrary $e$
• If you know $e$, $d$ and $n$ you can efficiently factor $n$, and obtain $p$ and $q$.

This means if you know one private key for a given $n$, you know all of them. Thus different persons should not share a modulus.

Such a scheme can be useful if one person needs multiple public keys. This person can use multiple $e$s with a shared $n$. There are few restrictions on the choice of $e$s, for example $e_3 = e_1 * e_2$ is broken.

Such key pairs are only useful in very specific scenarios. For example some blind signature schemes use different values of $e$ with shared $n$ to sign different currency denominations.

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