Is RSA secure if the private exponent is ``small''? Not if you send the same message to many people, even if you use different public bases for each message. This is called ``Hastad's broadcast attack''.
Suppose that the exponent is small, for example, . Let the private keys be , , , where each is a product of two large secret primes, and where the message satisfies . Assume that are pairwise relatively prime. Suppose that you send the same RSA encrypted message to people. The idea is very simple. Assume (mod ) is intercepted, as is (mod ) and also (mod ). (This assumption is a standard hypothesis when constructing an attack, since the encrypted messages are presumably traveling in the open). The Chinese remainder theorem allows one to determine (mod ). But since , , we must have . Thus, really is just . So take the cube root, and you have the message.