1. A private key generation apparatus intended for use in generating a private key corresponding to a user ID using the user ID as a public key, the apparatus comprising: an ID conversion unit configured to output a specific element value of a discrete logarithm cyclic group having a one-to-one correspondence relationship with an input user ID; and a private key calculation unit configured to: calculate a discrete logarithm result value based on the output specific element value; and calculate a private key having a one-to-one correspondence relationship with the input user ID.