Approximately counting semismooth integers

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.Comment: To appear in ISSAC 2013, Boston M

An Algorithm for Ennola's Second Theorem and Counting Smooth Numbers in Practice

Let $\Psi(x,y)$ count the number of positive integers $n\le x$ such that every prime divisor of $n$ is at most $y$. Given inputs $x$ and $y$, what is the best way to estimate $\Psi(x,y)$? We address this problem in three ways: with a new algorithm to estimate $\Psi(x,y)$, with a performance improvement to an established algorithm, and with empirically based advice on how to choose an algorithm to estimate $\Psi$ for the given inputs. Our new algorithm to estimate $\Psi(x,y)$ is based on Ennola's second theorem [Ennola69], which applies when $y0$. It takes $O(y^2/\log y)$ arithmetic operations of precomputation and $O(y\log y)$ operations per evaluation of $\Psi$. We show how to speed up Algorithm HT, which is based on the saddle-point method of Hildebrand and Tenenbaum [1986], by a factor proportional to $\log\log x$, by applying Newton's method in a new way. And finally we give our empirical advice based on five algorithms to compute estimates for $\Psi(x,y)$.The challenge here is that the boundaries of the ranges of applicability, as given in theorems, often include unknown constants or small values of $\epsilon>0$, for example, that cannot be programmed directly

Fast Bounds on the Distribution of Smooth Numbers

In this paper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for (x, y)

Divisibility, Smoothness and Cryptographic Applications

This paper deals with products of moderate-size primes, familiarly known as smooth numbers. Smooth numbers play a crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role