Mapping the discrete logarithm

Abstract

The discrete logarithm is a problem that surfaces frequently in the field of cryptography as a result of using the transformation $x\mapsto g^{x}modn$. Analysis of the security of many cryptographic algorithms depends on the assumption that it is statistically impossible to distinguish the use of this map from the use of a randomly chosen map with similar characteristics. This paper focuses on a prime modulus, $p$, for which it is shown that the basic structure of the functional graph produced by this map is largely dependent on an interaction between $g$ and $p-1$. We deal with two of the possible structures, permutations and binary functional graphs. Estimates exist for the shape of a random permutation, but similar estimates must be created for the binary functional graphs. Experimental data suggest that both the permutations and binary functional graphs correspond well to the theoretical predictions.