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 ↦→ gx mod n. 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.