A cryptographic system is described which is secure if and only if computing logarithms over GF(p) is infeasible. Previously published algorithms for computing this function require O(p^{1/2}) complexity in both time and space. An improved algorithm is derived which requires O =(\log^{2} p) complexity if p - 1 has only small prime factors. Such values of p must be avoided in the cryptosystem. Constructive uses for the new algorithm are also described.