A zero-knowledge proving system includes a proving mechanism for proving equality or inequality of two discrete logarithms and a verifying mechanism for verifying said equality or inequality. The proving mechanism stores public information including a designated operation scheme, two input numbers and , and two predetermined bases g and h, private information x which is a discrete logarithm of to the base g. After converting , and h to produce , and as follows: =^r; =^r; and =h^xr, the equality of a log and log and the equality of log_g and log_h are proved. The verifying mechanism verifies the equality of a log and log and the equality of log_g and log_h. Then, the received and are checked to determine the equality or inequality thereof, and it is determined whether the proof is acceptable, depending on the verification and the check results.