Pattern occurrence in the dyadic expansion of square root of two and an analysis of pseudorandom number generators

TL;DR

This paper investigates the relationship between pattern occurrences in the dyadic expansion of √2 and the distribution of accuracy parameters in pseudorandom number generators, revealing bounds and implications for cryptographic security.

Contribution

It establishes a theoretical link between dyadic pattern distributions in √2 and the occurrence of undesirable PRNG accuracy parameters, providing bounds and conjectures.

Findings

Asymptotic occurrence rate of patterns is bounded by zeroes' rate.

Classical conjecture implies at least 1/6 rate of undesirable parameters.

Examples of binary words achieve the theoretical bounds.

Abstract

Recently, designs of pseudorandom number generators (PRNGs) using integer-valued variants of logistic maps and their applications to some cryptographic schemes have been studied, due mostly to their ease of implementation and performance. However, it has been noted that this ease is reduced for some choices of the PRNGs accuracy parameters. In this article, we show that the distribution of such undesirable accuracy parameters is closely related to the occurrence of some patterns in the dyadic expansion of the square root of 2. We prove that for an arbitrary infinite binary word, the asymptotic occurrence rate of these patterns is bounded in terms of the asymptotic occurrence rate of zeroes. We also present examples of infinite binary words that tightly achieve the bounds. As a consequence, a classical conjecture on asymptotic evenness of occurrence of zeroes and ones in the dyadic expansion of the square root of 2 implies that the asymptotic rate of the undesirable accuracy parameters for the PRNGs is at least 1/6.