An Improved Algorithm for Computing Logarithms over GF(p) and Its Cryptographic Significance

Abstract

A cryptographic system is described which is secure if and only if computing log­ arithms 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 algo­ rithm 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.