Richard Ilemobade, Olufemi George, Jaiyeola Temitope Gbolahan
{"title":"(r,s,t)-逆拟群和环的通用性和同位素同构及其在密码学中的应用","authors":"Richard Ilemobade, Olufemi George, Jaiyeola Temitope Gbolahan","doi":"10.56415/qrs.v31.04","DOIUrl":null,"url":null,"abstract":"This paper introduced a condition called $\\mathcal{R}$-condition under which $(r,s,t)$-inverse quasigroups are universal. Middle isotopic $(r,s,t)$-inverse loops, satisfying the $\\mathcal{R}$-condition and possessing a trivial set of $r$-weak inverse permutations were shown to be isomorphic; isotopy-isomorphy for $(r,s,t)$-inverse loops. Isotopy-isomorphy for $(r,s,t)$-inverse loops was generally characterized. With the $\\mathcal{R}$-condition, it was shown that for positive integers $r$, $s$ and $t$, if there is a $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd(k,r+s+t)>1$, then there exists an $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd\\big(k(r+s+t), (r+s+t)^2\\big)$. The procedure of application of such $(r,s,t)$-inverse quasigroups to cryptography was described and explained, while the feasibility of such $(r,s,t)$-inverse quasigroups was illustrated with sample values of $k,r,s$ and $t$.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the universality and isotopy-isomorphy of (r,s,t)-inverse quasigroups and loops with applications to cryptography\",\"authors\":\"Richard Ilemobade, Olufemi George, Jaiyeola Temitope Gbolahan\",\"doi\":\"10.56415/qrs.v31.04\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper introduced a condition called $\\\\mathcal{R}$-condition under which $(r,s,t)$-inverse quasigroups are universal. Middle isotopic $(r,s,t)$-inverse loops, satisfying the $\\\\mathcal{R}$-condition and possessing a trivial set of $r$-weak inverse permutations were shown to be isomorphic; isotopy-isomorphy for $(r,s,t)$-inverse loops. Isotopy-isomorphy for $(r,s,t)$-inverse loops was generally characterized. With the $\\\\mathcal{R}$-condition, it was shown that for positive integers $r$, $s$ and $t$, if there is a $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd(k,r+s+t)>1$, then there exists an $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd\\\\big(k(r+s+t), (r+s+t)^2\\\\big)$. The procedure of application of such $(r,s,t)$-inverse quasigroups to cryptography was described and explained, while the feasibility of such $(r,s,t)$-inverse quasigroups was illustrated with sample values of $k,r,s$ and $t$.\",\"PeriodicalId\":38681,\"journal\":{\"name\":\"Quasigroups and Related Systems\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quasigroups and Related Systems\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.56415/qrs.v31.04\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quasigroups and Related Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.56415/qrs.v31.04","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
On the universality and isotopy-isomorphy of (r,s,t)-inverse quasigroups and loops with applications to cryptography
This paper introduced a condition called $\mathcal{R}$-condition under which $(r,s,t)$-inverse quasigroups are universal. Middle isotopic $(r,s,t)$-inverse loops, satisfying the $\mathcal{R}$-condition and possessing a trivial set of $r$-weak inverse permutations were shown to be isomorphic; isotopy-isomorphy for $(r,s,t)$-inverse loops. Isotopy-isomorphy for $(r,s,t)$-inverse loops was generally characterized. With the $\mathcal{R}$-condition, it was shown that for positive integers $r$, $s$ and $t$, if there is a $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd(k,r+s+t)>1$, then there exists an $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd\big(k(r+s+t), (r+s+t)^2\big)$. The procedure of application of such $(r,s,t)$-inverse quasigroups to cryptography was described and explained, while the feasibility of such $(r,s,t)$-inverse quasigroups was illustrated with sample values of $k,r,s$ and $t$.