To construct a Digital Signature, public key encryption systems must be in place. The construction of the Digital Signature entails applying a hash function to the message by concatenation of the message with a known secret key and then applying a mathematical function which will produce a fixed length output known as the digest. The digest is then encrypted with the public decryption key which produces a signature that can be appended to the message to verify that the message is from the genuine source.
The receiver recalculates the hash function and compared with the signature after applying the public key. If the two match, then because only the originator could have known the hash function and the private key, the message must be genuine.