Performance Comparison of Linear Sieve and Cubic Sieve Algorithms for Discrete Logarithms over Prime Fields

It is of interest in cryptographic applications to obtain practical performance improvements for the discrete logarithm problem over prime fields Fp with p of size ≤ 500 bits. The linear sieve and the cubic sieve methods described in Coppersmith, Odlyzko and Schroeppel’s paper [3] are two practical algorithms for computing discrete logarithms over prime fields. The cubic sieve algorithm is asymptotically faster than the linear sieve algorithm. We discuss an efficient implementation of the cubic sieve algorithm incorporating two heuristic principles. We demonstrate through empirical performance measures that for a special class of primes the cubic sieve method runs about two to three times faster than the linear sieve method even in cases of small prime fields of size about 150 bits.