Since the public key is derived from the private key and are both mathematically related, the public key can be used to verify signatures created using the private key. The private key can also decrypt messages encrypted using the public key. For example, Alice can encrypt a message to Bob using Bob's public key if she does not want to reveal the message to a third-party. Bob will be able to decrypt Alice's messages using his private key and since only Bob knows the private key, only he can decrypt the message.