This is a slight reversal from the norm but...
Suppose Alice wants to ensure that everyone knows that a message will be from her. Instead of giving out the encryption key, she broadcasts the decryption key, and encrypts her messages privately. She doesn't care that anybody can read her messages, only that someone can't imitate her.
So if I want to imitate Alice, and all I have is a decryption key and the message, is it possible to fake a message from Alice?
Edit: so to clarify, we'll assume a literate message is valid. We'll also add the constraint that the attacker wants to mimic a specific Message (M), not just any message, or previous messages will do.