Counting RSA-integers

TL;DR

This paper proves the asymptotic count of RSA-integers, integers of the form n=pq with primes p and q of comparable size, and determines the exact constant involved in their distribution.

Contribution

It rigorously establishes the asymptotic formula for RSA-integers and explicitly computes the constant c_r as 2 log r, confirming a folklore result.

Findings

Asymptotic formula for RSA-integers count established

Constant c_r computed as 2 log r

Validates folklore in cryptography literature

Abstract

In the RSA cryptosystem integers of the form n=p.q with p and q primes of comparable size (`RSA-integers') play an important role. It is a folklore result of cryptographers that C_r(x), the number of integers n<=x that are of the form n=pq with p and q primes such that p<q<rp, is for fixed r>1 asymptotically equal to c_r*x*log^{-2}x for some constant c_r>0. Here we prove this and show that c_r=2log r.