{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T16:55:50Z","timestamp":1761238550206,"version":"build-2065373602"},"reference-count":7,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,8,28]],"date-time":"2018-08-28T00:00:00Z","timestamp":1535414400000},"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":"<jats:p>RSA key pairs are normally generated from two large primes p and q. We consider what happens if they are generated from two integers s and r, where r is prime, but unbeknownst to the user, s is not. Under most circumstances, the correctness of encryption and decryption depends on the choice of the public and private exponents e and d. In some cases, specific     ( s , r )     pairs can be found for which encryption and decryption will be correct for any     ( e , d )     exponent pair. Certain s exist, however, for which encryption and decryption are correct for any odd prime     r \u2224 s    . We give necessary and sufficient conditions for s with this property.<\/jats:p>","DOI":"10.3390\/info9090216","type":"journal-article","created":{"date-parts":[[2018,8,28]],"date-time":"2018-08-28T12:19:06Z","timestamp":1535458746000},"page":"216","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Composite Numbers That Give Valid RSA Key Pairs for Any Coprime p"],"prefix":"10.3390","volume":"9","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":[[2018,8,28]]},"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","unstructured":"Hinek, M. (2010). Cryptanalysis of RSA and Its Variants, CRC Press.","DOI":"10.1201\/9781420075199"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/0022-314X(80)90084-0","article-title":"Probabalistic Algorithm for Testing Primality","volume":"12","author":"Rabin","year":"1980","journal-title":"J. Number Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1090\/S0002-9904-1939-06953-X","article-title":"On Fermat\u2019s Simple Theorem","volume":"45","author":"Chernick","year":"1935","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"703","DOI":"10.2307\/2118576","article-title":"There are Infinitely Many Carmichael Numbers","volume":"140","author":"Alford","year":"1994","journal-title":"Ann. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02403921","article-title":"Some Problems of \u2018Partitio Numerorum\u2019. III. On the Expression of a Number as a Sum of Primes","volume":"44","author":"Hardy","year":"1923","journal-title":"Acta Math."},{"key":"ref_7","first-page":"02.2.1","article-title":"Carmichael Numbers of the Form (6m + 1)(12m + 1)(18m + 1)","volume":"5","author":"Dubner","year":"2002","journal-title":"J. Integer Seq."}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/9\/9\/216\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:21:35Z","timestamp":1760196095000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/9\/9\/216"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,28]]},"references-count":7,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2018,9]]}},"alternative-id":["info9090216"],"URL":"https:\/\/doi.org\/10.3390\/info9090216","relation":{},"ISSN":["2078-2489"],"issn-type":[{"type":"electronic","value":"2078-2489"}],"subject":[],"published":{"date-parts":[[2018,8,28]]}}}