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 one discrete logarithm cyclic group having a one-to-one correspondence relationship with an input user ID by using a value of a previously set length irrespective of a length of the input user ID; and a private key calculation unit configured to: calculate a discrete logarithm result value based on the output specific element value of the one discrete logarithm cyclic group; and calculate a private key having a one-to-one correspondence relationship with the input user ID by using parameters set so that the input user ID as the input value and the discrete logarithm result value as the output value are in one-to-one correspondence to each other, wherein the private key calculation unit comprises: a setting unit configured to set parameters for one-to-one correspondence between the input user ID and the discrete logarithm result value; a pre-computation table generation unit configured to store a value of a result of performing a multiplication operation using the set parameters in the form of a data structure to generate a lookup table, and store a chain value of a cyclic function value taking into account a time-memory trade off using the lookup table, in the form of a data structure, to generate a pre-computation table; and a discrete logarithm calculation unit configured to receive data from the setting unit and the pre-computation table generation unit, and calculate a discrete logarithm result value using an index value of the pre-computation table, wherein at least one of the ID conversion unit and the private key calculation unit is implemented as a hardware component.