Although Alice could have sent a private message to the bank, signed it, and ensured the integrity of the message, she still needs to be sure that she is really communicating with the bank. This means that she needs to be sure that the public key she is using corresponds to the bank's private key. Similarly, the bank also needs to verify that the message signature really corresponds to Alice's signature.

If each party has a certificate which validates the other's identity, confirms the public key, and is signed by a trusted agency, then they both are assured that they are communicating with whom they think they are. Each party uses the public key of the trusted agency to verify the certificate of the other party and subsequently to ensure the authenticity of the users' public key.