Integers with a large smooth divisor

TL;DR

This paper investigates the counting function for integers with large smooth divisors and explores cryptographic applications of these findings.

Contribution

It introduces a new function $ heta(x,y,z)$ for counting integers with specific divisor properties and discusses their cryptographic relevance.

Findings

Derived asymptotic estimates for $ heta(x,y,z)$

Identified cryptographic implications of large smooth divisors

Provided bounds for the distribution of such integers

Abstract

We study the function $\Theta(x,y,z)$ that counts the number of positive integers $n\le x$ which have a divisor $d>z$ with the property that $p\le y$ for every prime $p$ dividing $d$. We also indicate some cryptographic applications of our results.