{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T16:21:38Z","timestamp":1776442898644,"version":"3.51.2"},"reference-count":56,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T00:00:00Z","timestamp":1704067200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,9,28]]},"abstract":"Abstract<\/jats:title>\n \n Let\n \n \n \n \n N<\/m:mi>\n =<\/m:mo>\n p<\/m:mi>\n q<\/m:mi>\n <\/m:math>\n N=pq<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be the product of two balanced prime numbers\n \n \n \n \n p<\/m:mi>\n <\/m:math>\n p<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n \n \n q<\/m:mi>\n <\/m:math>\n q<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In Elkamchouchi et al. (\n Extended RSA cryptosystem and digital signature schemes in the domain of Gaussian integers<\/jats:italic>\n . In: ICCS 2002. vol. 1. IEEE Computer Society; 2002. p. 91\u20135.) introduced an Rivest-Shamir-Adleman (RSA)-like cryptosystem that uses the key equation\n \n \n \n \n e<\/m:mi>\n d<\/m:mi>\n \u2212<\/m:mo>\n k<\/m:mi>\n \n (<\/m:mo>\n \n \n \n p<\/m:mi>\n <\/m:mrow>\n \n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \n (<\/m:mo>\n \n \n \n q<\/m:mi>\n <\/m:mrow>\n \n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:math>\n ed-k\\left({p}^{2}-1)\\left({q}^{2}-1)=1<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , instead of the classical RSA key equation\n \n \n \n \n e<\/m:mi>\n d<\/m:mi>\n \u2212<\/m:mo>\n k<\/m:mi>\n \n (<\/m:mo>\n \n p<\/m:mi>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \n (<\/m:mo>\n \n q<\/m:mi>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:math>\n ed-k\\left(p-1)\\left(q-1)=1<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Another variant of RSA, presented in Murru and Saettone (\n A novel RSA-like cryptosystem based on a generalization of the R\u00e9dei rational functions<\/jats:italic>\n . In: NuTMiC 2017. vol. 10737 of Lecture Notes in Computer Science. Springer; 2017. p. 91\u2013103), uses the key equation\n \n \n \n \n e<\/m:mi>\n d<\/m:mi>\n \u2212<\/m:mo>\n k<\/m:mi>\n \n (<\/m:mo>\n \n \n \n p<\/m:mi>\n <\/m:mrow>\n \n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n +<\/m:mo>\n p<\/m:mi>\n +<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \n (<\/m:mo>\n \n \n \n q<\/m:mi>\n <\/m:mrow>\n \n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n +<\/m:mo>\n q<\/m:mi>\n +<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:math>\n ed-k\\left({p}^{2}+p+1)\\left({q}^{2}+q+1)=1<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Despite the authors\u2019 claims of enhanced security, both schemes remain vulnerable to adaptations of common RSA attacks. Let\n \n \n \n \n n<\/m:mi>\n <\/m:math>\n n<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be an integer. This article proposes two families of RSA-like encryption schemes: one employs the key equation\n \n \n \n \n e<\/m:mi>\n d<\/m:mi>\n \u2212<\/m:mo>\n k<\/m:mi>\n \n (<\/m:mo>\n \n \n \n p<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \n (<\/m:mo>\n \n \n \n q<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:math>\n ed-k\\left({p}^{n}-1)\\left({q}^{n}-1)=1<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for\n \n \n \n \n n<\/m:mi>\n ><\/m:mo>\n 0<\/m:mn>\n <\/m:math>\n n\\gt 0<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , while the other uses\n \n \n \n \n e<\/m:mi>\n d<\/m:mi>\n \u2212<\/m:mo>\n k<\/m:mi>\n \n [<\/m:mo>\n \n \n (<\/m:mo>\n \n \n \n p<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \n (<\/m:mo>\n \n \n \n q<\/m:mi>\n <\/m:mrow>\n \n n<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n ]<\/m:mo>\n <\/m:mrow>\n \u2044<\/m:mo>\n \n [<\/m:mo>\n \n \n (<\/m:mo>\n \n p<\/m:mi>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \n (<\/m:mo>\n \n q<\/m:mi>\n \u2212<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n ]<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:math>\n ed-k\\left[\\left({p}^{n}-1)\\left({q}^{n}-1)]\/\\left[\\left(p-1)\\left(q-1)]=1<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for\n \n \n \n \n n<\/m:mi>\n ><\/m:mo>\n 1<\/m:mn>\n <\/m:math>\n n\\gt 1<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Note that we remove the conventional assumption of primes having equal bit sizes. In this scenario, we show that regardless of the choice of\n \n \n \n \n n<\/m:mi>\n <\/m:math>\n n<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , continued fraction-based attacks can still recover the secret exponent. Additionally, this work fills a gap in the literature by establishing an equivalent of Wiener\u2019s attack when the primes do not have the same bit size.\n <\/jats:p>","DOI":"10.1515\/jmc-2024-0013","type":"journal-article","created":{"date-parts":[[2024,9,28]],"date-time":"2024-09-28T04:22:51Z","timestamp":1727497371000},"source":"Crossref","is-referenced-by-count":5,"title":["A security analysis of two classes of RSA-like cryptosystems"],"prefix":"10.1515","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2169-2264","authenticated-orcid":false,"given":"Paul","family":"Cotan","sequence":"first","affiliation":[{"name":"Simion Stoilow Institute of Mathematics of the Romanian Academy , 21 Calea Grivitei , Bucharest , Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3953-2744","authenticated-orcid":false,"given":"George","family":"Te\u015feleanu","sequence":"additional","affiliation":[{"name":"Simion Stoilow Institute of Mathematics of the Romanian Academy , 21 Calea Grivitei , Bucharest , Romania"}]}],"member":"374","published-online":{"date-parts":[[2024,9,28]]},"reference":[{"key":"2025120600251248577_j_jmc-2024-0013_ref_001","doi-asserted-by":"crossref","unstructured":"Rivest RL, Shamir A, Adleman L. 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