Cryptographic Hash Functions

Hash functions are only useful when there is a secure way of sending the hash value.

For example, Bob can publish a hash of his public key on his business card
Putting the whole key on there would be too big.
But Alice can download Bob's key from the Internet, hash it herself, and verify that the hash matches the one on Bob's card.

You can't just send an unencrypted message and hash to get integrity assurance
Even if you don't care about secrecy!
Mallory can just change the message, and just compute the new hash value himself.