Learning public-private key systems

Signatures in public-private key systems calculate a number so that that number, s, when put into a prover function, outputs the message that was signed, m. The proof that underlies the signature is that without knowing the secret key d, it is very hard to use generate a number that when put into the prover outputs m.

In RSA, the signature formula is s = m^d mod n, and verification is m = s^e mod n, where d is the private key and e and n the public key. The signature does the mathematical inverse of what the verification does, just as decryption is done using the mathematical inverse of encryption, signature and decryption are the same function, and verification and encryption the same function.