On the cubic sieve method for computing discrete logarithms over prime fields

In this paper, we report efficient implementations of the linear sieve and the cubic sieve methods for computing discrete logarithms over prime fields. We demonstrate through empirical performance measures that for a special class of primes the cubic sieve method runs about two times faster than the linear sieve method even in cases of small prime fields of the size about 150 bits. We also provide a heuristic estimate of the number of solutions of the congruence X3 ≡ Y 2Z (mod p) that is of central importance in the cubic sieve method.