On RSA Moduli with Almost Half of the Bits Prescribed

TL;DR

This paper improves the ability to find RSA moduli with specific bit patterns by leveraging character sum estimates, allowing for more prescribed bits and removing reliance on unproven hypotheses.

Contribution

It applies Iwaniec's character sum estimates to enhance results on RSA moduli with prescribed bits, achieving more bits specified and providing unconditional results.

Findings

Can specify about n bits instead of n/2 bits

Improves distribution results of RSA moduli with prescribed bits

Provides unconditional version of a combinatorial result

Abstract

We show that using character sum estimates due to H. Iwaniec leads to an improvement of recent results about the distribution and finding RSA moduli $M=pl$, where $p$ and $l$ are primes, with prescribed bit patterns. We are now able to specify about $n$ bits instead of about $n/2$ bits as in the previous work. We also show that the same result of H. Iwaniec can be used to obtain an unconditional version of a combinatorial result of W. de Launey and D. Gordon that was originally derived under the Extended Riemann Hypothesis.