Next Article in Journal
A Novel Approach for Web Service Recommendation Based on Advanced Trust Relationships
Previous Article in Journal
Assisting Forensic Identification through Unsupervised Information Extraction of Free Text Autopsy Reports: The Disappearances Cases during the Brazilian Military Dictatorship
Article Menu
Issue 7 (July) cover image

Export Article

Open AccessArticle

Idempotent Factorizations of Square-Free Integers

Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA
Information 2019, 10(7), 232; https://doi.org/10.3390/info10070232
Received: 20 June 2019 / Accepted: 3 July 2019 / Published: 6 July 2019
  |  
PDF [311 KB, uploaded 6 July 2019]
  |  

Abstract

We explore the class of positive integers n that admit idempotent factorizations n = p ¯ q ¯ such that λ ( n ) ( p ¯ 1 ) ( q ¯ 1 ) , where λ is the Carmichael lambda function. Idempotent factorizations with p ¯ and q ¯ prime have received the most attention due to their cryptographic advantages, but there are an infinite number of n with idempotent factorizations containing composite p ¯ and/or q ¯ . Idempotent factorizations are exactly those p ¯ and q ¯ that generate correctly functioning keys in the Rivest–Shamir–Adleman (RSA) 2-prime protocol with n as the modulus. While the resulting p ¯ and q ¯ have no cryptographic utility and therefore should never be employed in that capacity, idempotent factorizations warrant study in their own right as they live at the intersection of multiple hard problems in computer science and number theory. We present some analytical results here. We also demonstrate the existence of maximally idempotent integers, those n for which all bipartite factorizations are idempotent. We show how to construct them, and present preliminary results on their distribution. View Full-Text
Keywords: cryptography; abstract algebra; Rivest–Shamir–Adleman (RSA); computer science education; cryptography education; number theory; factorization cryptography; abstract algebra; Rivest–Shamir–Adleman (RSA); computer science education; cryptography education; number theory; factorization
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Fagin, B. Idempotent Factorizations of Square-Free Integers. Information 2019, 10, 232.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Information EISSN 2078-2489 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top