Common exponent attack

Suppose we're sending the same message to Alice, Bob, and Carol, and they all have the same small exponent.

Message to Alice:

Message to Bob:

Message to Carol:

Eve (an eavesdropper) hears the messages.  Eve knows

and similarly for the second half of the message.  (Everything here except P is public information!)  Then the Chinese Remainder Theorem (c. 1st century C.E.) says Eve can recover

using the magic formula

But now Eve can use the small message attack:

This is guaranteed to work if there are at least e messages.  Moral:  don't send identical messages!

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