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 → g x 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.

Statistics for fixed points of the self-power map

“…Our first attempt assumed that F (p) was normally distributed around the predicted value d|p−1 φ(d) d . (The normality assumption had been successfully used for the discrete exponential map in, e.g., [3], see also [17]. Furthermore, it appeared to be justified by the Central Limit Theorem given the number of primes we were intending to test.…”

Artin's Primitive Root Conjecture – A Survey

“…If S is a finite set and f : S → S a mapping, then one can associate a functional graph to the mapping f by interpreting each element in S as a vertex. The edges are defined such that an edge (a, b) is in the graph iff f (a) = b. Cloutier and Holden [91] considered the functional graph associated to the mapping x → g x modulo p, with p a prime and S = {1, 2, . .…”

Applications of Number Theory in Statistics

“…There are applications of this problem in cryptography which is the subject of sending messages in a secret way, ensuring the security of the information (see for e.g., A.M.Odlyzko [6]). The problem of mapping the discrete logarithm has been considered by D.Cloutier and J.Holden [2]. The structure in the discrete logarithm has been studied by A.Hoffman [4].…”

“…We say that r is a primitive root modulo p if . Let r be any primitive root modulo p and g D. Cloutier and J. Holden [2] that the values of g that produce an m-ary graph are precisely those for which gcd (α, p-1) = m.…”