Actually, there is one additional condition required for public key cryptography. There must also be no computationally feasible way of deriving the private key from the public key. The reasons are straightforward:

		    Let k-priv be the "private key."
		    Let k-pub be the "public key."
		    Let X() be a computationally feasible
		        transformation of any public key into a private key.
		    Let D{k-priv}() be the decryption function
		        corresponding to encryption function E{k-pub}().
		    By definition,
		        M = D{k-priv}(E{k-pub}(M))
		    We may define, trivially,
		        D'{k-pub} = D{X(k-pub)}() = D{k-priv}()
		    Therefore,
		        M = D'{k-pub}(E{k-pub}(M))
		    By using D'(), we have reduced the protocol to standard
		        symmetric encryption!
		    

The computational feasibility question is important. If derivation of the private key from the public key is possible, but not feasible, then we can decrypt using the public key in mathematical abstraction, but we cannot get it done in the real world.

The radical result of Diffie's and Hellman's idea is a class of algorithms where we can tell the whole world a public key to use, but rest easy knowing that upon encryption of a message with this public key, only the holders of the private key can decrypt it. We can send secret messages without needing to share secrets (that is, using a key) with our correspondents.