{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:25:22Z","timestamp":1772252722977,"version":"3.50.1"},"reference-count":11,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,7,6]],"date-time":"2019-07-06T00:00:00Z","timestamp":1562371200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"We explore the class of positive integers n that admit idempotent factorizations n = p \u00af q \u00af such that \u03bb ( n ) \u2223 ( p \u00af \u2212 1 ) ( q \u00af \u2212 1 ) , where \u03bb is the Carmichael lambda function. Idempotent factorizations with p \u00af and q \u00af 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 \u00af and\/or q \u00af . Idempotent factorizations are exactly those p \u00af and q \u00af that generate correctly functioning keys in the Rivest\u2013Shamir\u2013Adleman (RSA) 2-prime protocol with n as the modulus. While the resulting p \u00af and q \u00af 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.<\/jats:p>","DOI":"10.3390\/info10070232","type":"journal-article","created":{"date-parts":[[2019,7,8]],"date-time":"2019-07-08T03:01:31Z","timestamp":1562554891000},"page":"232","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Idempotent Factorizations of Square-Free Integers"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5940-2598","authenticated-orcid":false,"given":"Barry","family":"Fagin","sequence":"first","affiliation":[{"name":"Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1145\/359340.359342","article-title":"A Method for Obtaining Digital Signatures and Public-Key Cryptosystem","volume":"21","author":"Rivest","year":"1978","journal-title":"Commun. ACM"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1090\/S0002-9904-1910-01892-9","article-title":"Note on a New Number Theory Function","volume":"16","author":"Carmichael","year":"1910","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/0893-9659(88)90081-X","article-title":"On Using Primes for Public Key Encryption Systems","volume":"1","author":"Huthnance","year":"1988","journal-title":"Appl. Math. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Pinch, R.G.E. (1997, January 17\u201319). On Using Carmichael Numbers for Public Key Encryption Systems. Proceedings of the IMA International Conference on Cryptography and Coding, Cirencester, UK.","DOI":"10.1007\/BFb0024472"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Fagin, B. (2018). Composite Numbers That Give Valid RSA Key Pairs For Any Coprime p. Information, 9.","DOI":"10.3390\/info9090216"},{"key":"ref_6","unstructured":"Pinch, R.G.E. (2007, January 8\u201311). The Carmichael Numbers up to 1021. Proceedings of the Conference on Algorithmic Number Theory, Turku, Finland."},{"key":"ref_7","unstructured":"Fagin, B., and OEIS Foundation Inc. (2019, July 04). The On-Line Encyclopedia of Integer Sequences. Squarefree n with \u22653 factors That Admit Idempotent Factorizations n = p\u00afq\u00af. Available online: http:\/\/oeis.org\/A306330."},{"key":"ref_8","unstructured":"Fagin, B., and OEIS Foundation Inc. (2019, July 04). The On-Line Encyclopedia of Integer Sequences. Squarefree n with Fully Composite Idempotent Factorizations. Available online: http:\/\/oeis.org\/A306508."},{"key":"ref_9","unstructured":"Fagin, B., and OEIS Foundation Inc. (2019, July 04). The On-Line Encyclopedia of Integer Sequences. Maximally Idempotent Integers with \u22653 Factors. Available online: http:\/\/oeis.org\/A306812."},{"key":"ref_10","unstructured":"Fagin, B., and OEIS Foundation Inc. (2019, July 04). The On-Line Encyclopedia of Integer Sequences. \u201cStrong Impostors\u201d \u2260 0 (mod 4). Available online: http:\/\/oeis.org\/A318555."},{"key":"ref_11","unstructured":"Fagin, B. (March, January 27). Teaching RSA: What Happens When One of Your Primes Isn\u2019t?. Proceedings of the 50th ACM Technical Symposium on Computer Science Education (SIGCSE \u201919), Minneapolis, MN, USA."}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/10\/7\/232\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:03:15Z","timestamp":1760187795000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/10\/7\/232"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,6]]},"references-count":11,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2019,7]]}},"alternative-id":["info10070232"],"URL":"https:\/\/doi.org\/10.3390\/info10070232","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201906.0208.v1","asserted-by":"object"}]},"ISSN":["2078-2489"],"issn-type":[{"value":"2078-2489","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,7,6]]}}}