Vol. 3, No. 2, 2010

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Mapping the discrete logarithm

Daniel Cloutier and Joshua Holden

Vol. 3 (2010), No. 2, 197–213
Abstract

The discrete logarithm is a problem that surfaces frequently in the field of cryptography as a result of using the transformation $x↦{g}^{x}\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}n$. 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.

Keywords
discrete logarithm problem, random map, functional graph
Mathematical Subject Classification 2000
Primary: 11Y16
Secondary: 11-04, 94A60, 05A15