Broadcasting Private Message Securely

Wireless communication channels are easier to eavesdrop and harder to secure – even towards unintentional eavesdrop- pers. As an example consider a sender, Alice, who wants to send private messages to multiple (say three) receivers, Bob, Calvin and David, within her transmission radius, and assume public feedback from the receivers to Alice. When Alice broadcasts a message W1 intended for Bob, Calvin and David should also try to overhear, as the side information they possibly collect can enable Alice to make her following broadcast transmissions more efficient; but then, this collected side information would allow Calvin and David to learn parts of Bob’s message. Even worse, Calvin and David could try to put together the parts they overheard, to extract increased information about Bob’s message. Can we, in such a setting, keep the message for each user information theoretically secure from the other users, even if these collaborate? Moreover, can we do so, when the users can only communicate through shared wireless broadcast channels?