Approximately counting semismooth integers

TL;DR

This paper investigates algorithms to estimate the count of semismooth integers, which are crucial in integer factoring algorithms like the number field sieve, by generalizing Buchstab's identity for better parameter setting.

Contribution

It introduces several algorithms based on a generalized Buchstab's identity to approximate the count of semismooth integers up to a given bound.

Findings

Algorithms provide accurate estimates of semismooth integers.

Generalized Buchstab's identity enhances approximation methods.

Potential to optimize factoring algorithm parameters.

Abstract

An integer $n$ is $(y,z)$-semismooth if $n=pm$ where $m$ is an integer with all prime divisors $\le y$ and $p$ is 1 or a prime $\le z$. arge quantities of semismooth integers are utilized in modern integer factoring algorithms, such as the number field sieve, that incorporate the so-called large prime variant. Thus, it is useful for factoring practitioners to be able to estimate the value of $\Psi(x,y,z)$, the number of $(y,z)$-semismooth integers up to $x$, so that they can better set algorithm parameters and minimize running times, which could be weeks or months on a cluster supercomputer. In this paper, we explore several algorithms to approximate $\Psi(x,y,z)$ using a generalization of Buchstab's identity with numeric integration.