Statistical Analysis of Binary Functional Graphs of the Discrete Logarithm

TL;DR

This paper analyzes the statistical properties of graphs derived from the discrete logarithm to assess cryptographic security, revealing unexpected distribution patterns and providing theoretical asymptotic distributions.

Contribution

It introduces the first detailed statistical analysis of discrete logarithm graphs, including expected values, variances, and asymptotic distributions, challenging initial Gaussian assumptions.

Findings

Expected value and variance of graph properties calculated

Experimental data did not follow Gaussian distribution

Theoretical asymptotic distributions derived

Abstract

The increased use of cryptography to protect our personal information makes us want to understand the security of cryptosystems. The security of many cryptosystems relies on solving the discrete logarithm, which is thought to be relatively difficult. Therefore, we focus on the statistical analysis of certain properties of the graph of the discrete logarithm. We discovered the expected value and variance of a certain property of the graph and compared the expected value to experimental data. Our finding did not coincide with our intuition of the data following a Gaussian distribution given a large sample size. Thus, we found the theoretical asymptotic distributions of certain properties of the graph.