{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T09:43:59Z","timestamp":1779183839877,"version":"3.51.4"},"reference-count":23,"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":[[2024,11,26]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Given that the tropical Stickel protocol and its variants are all vulnerable to the generalized Kotov\u2013Ushakov attack, we suggest employing the max-min semiring and, more generally, max-\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2024-0014_eq_001.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mi>T<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>T<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    semiring where the multiplication is based on a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2024-0014_eq_002.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mi>T<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>T<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -norm, as a framework to implement the Stickel protocol. While the Stickel protocol over max-min semiring or max-\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2024-0014_eq_003.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mi>T<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>T<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    semiring remains susceptible to a form of Kotov\u2013Ushakov attack, we demonstrate that it exhibits significantly increased resistance against this attack when compared to the tropical (max-plus) implementation.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2024-0014","type":"journal-article","created":{"date-parts":[[2024,11,26]],"date-time":"2024-11-26T11:00:23Z","timestamp":1732618823000},"source":"Crossref","is-referenced-by-count":5,"title":["On implementation of Stickel's key exchange protocol over max-min and max-\n                    <i>T<\/i>\n                    semirings"],"prefix":"10.1515","volume":"18","author":[{"given":"Sulaiman","family":"Alhussaini","sequence":"first","affiliation":[{"name":"School of Mathematics , University of Birmingham , Birmingham , Edgbaston B15 2TT , United Kingdom"}]},{"given":"Serge\u012d","family":"Sergeev","sequence":"additional","affiliation":[{"name":"School of Mathematics , University of Birmingham , Birmingham , Edgbaston B15 2TT , United Kingdom"}]}],"member":"374","published-online":{"date-parts":[[2024,11,26]]},"reference":[{"key":"2025120600251213020_j_jmc-2024-0014_ref_001","doi-asserted-by":"crossref","unstructured":"Diffie W, Hellman M. New directions in cryptography. IEEE Trans Inform Theory. 1976;22(6):644\u201354.","DOI":"10.1109\/TIT.1976.1055638"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_002","doi-asserted-by":"crossref","unstructured":"Stickel E. A new method for exchanging secret keys. In: Third International Conference on Information Technology and Applications (ICITA\u201905). Vol. 2; 2005. p. 426\u201330.","DOI":"10.1109\/ICITA.2005.33"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_003","doi-asserted-by":"crossref","unstructured":"Grigoriev D, Shpilrain V. Tropical cryptography. Commun Algebra. 2013;42:2624\u201332. https:\/\/api.semanticscholar.org\/CorpusID:6744219.","DOI":"10.1080\/00927872.2013.766827"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_004","doi-asserted-by":"crossref","unstructured":"Kotov M, Ushakov A. Analysis of a key exchange protocol based on tropical matrix algebra. J Math Cryptol. 2018;12(3):137\u201341. https:\/\/doi.org\/10.1515\/jmc-2016-0064, [cited 2023-11-26].","DOI":"10.1515\/jmc-2016-0064"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_005","doi-asserted-by":"crossref","unstructured":"Muanalifah A, Sergeev S. Modifying the tropical version of Stickel\u2019s key exchange protocol. Appl Math. 2020 Dec;65:727\u201353.","DOI":"10.21136\/AM.2020.0325-19"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_006","doi-asserted-by":"crossref","unstructured":"Grigoriev D, Shpilrain V. Tropical cryptography II: Extensions by homomorphisms. Commun Algebra. 2019;47(10):4224\u20139. 10.1080\/00927872.2019.1581213.","DOI":"10.1080\/00927872.2019.1581213"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_007","doi-asserted-by":"crossref","unstructured":"Isaac S, Kahrobaei D. A closer look at the tropical cryptography. Int J Comput Math Comput Syst Theory. 2021;6(2):137\u201342. 10.1080\/23799927.2020.1862303.","DOI":"10.1080\/23799927.2020.1862303"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_008","doi-asserted-by":"crossref","unstructured":"Muanalifah A, Sergeev S. On the tropical discrete logarithm problem and security of a protocol based on tropical semidirect product. Commun Algebra. 2022;50(2):861\u201379. 10.1080\/00927872.2021.1975125.","DOI":"10.1080\/00927872.2021.1975125"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_009","doi-asserted-by":"crossref","unstructured":"Rudy D, Monico C. Remarks on a tropical key exchange system. J Math Cryptol. 2020;15(1):280\u20133.","DOI":"10.1515\/jmc-2019-0061"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_010","doi-asserted-by":"crossref","unstructured":"Nitica V, Sergeev S. Tropical convexity over max-min semiring. In: Litvinov GL, Sergeev SN, editors. Tropical and idempotent mathematics and applications. Vol. 616 of Contemporary Mathematics. Providence, RI, USA: American Mathematical Society; 2014. p. 241\u201360.","DOI":"10.1090\/conm\/616\/12301"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_011","doi-asserted-by":"crossref","unstructured":"Gavalec M, N\u011bmcov\u00e1 Z, Sergeev S. Tropical linear algebra with the \u0141ukasiewicz T-norm. Fuzzy Sets Syst. 2015;276:131\u201348. Theme: Logic and Algebra. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0165011414005119.","DOI":"10.1016\/j.fss.2014.11.008"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_012","unstructured":"Di Nola A, Pedrycz W, Sessa S. Fuzzy relation equations under LSC and USC T-norms and their Boolean solutions. Stochastica. 1987;11(2\u20133):151\u201383."},{"key":"2025120600251213020_j_jmc-2024-0014_ref_013","doi-asserted-by":"crossref","unstructured":"Higashi M, Klir GJ. Resolution of finite fuzzy relation equations. Fuzzy Sets Syst. 1984;13:65\u201382.","DOI":"10.1016\/0165-0114(84)90026-5"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_014","unstructured":"Klir GJ, Yuan B. Fuzzy sets fuzzy logic. Theory and Applications. Saddle River, NJ, United States: Prentice Hall; 1995."},{"key":"2025120600251213020_j_jmc-2024-0014_ref_015","doi-asserted-by":"crossref","unstructured":"Di Nola A, Lettieri A. Finite BL algebras. Discrete Math. 2003;269:93\u2013112.","DOI":"10.1016\/S0012-365X(02)00754-9"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_016","doi-asserted-by":"crossref","unstructured":"Huang H, Li C, Deng L. Public-key cryptography based on tropical circular matrices. Appl Sci. 2022;12(15):7401. https:\/\/www.mdpi.com\/2076-3417\/12\/15\/7401.","DOI":"10.3390\/app12157401"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_017","doi-asserted-by":"crossref","unstructured":"Amutha B, Perumal R. Key exchange protocols based on tropical circulant and anti-circulant matrices. AIMS Math. 2023;8(7):17304\u201334.","DOI":"10.3934\/math.2023885"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_018","unstructured":"Collett C. Public key cryptography in max-plus algebra. MSc Dissertation. Birmingham, UK: University of Birmingham; School of Mathematics; 2023."},{"key":"2025120600251213020_j_jmc-2024-0014_ref_019","doi-asserted-by":"crossref","unstructured":"Gavalec M. Solvability and unique solvability of max-min fuzzy equations. Fuzzy Sets Syst. 2001;124(3):385\u201393. Fuzzy logic. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0165011401001087.","DOI":"10.1016\/S0165-0114(01)00108-7"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_020","unstructured":"Zahariev Z. Solving max-min fuzzy linear systems of equations. Algorithm and software. Annual of \u201cInformatics\u201d section Union of Scientists in Bulgaria. 2013;6:1\u201316. http:\/\/e-university.tu-sofia.bg\/e-publ\/files\/12485_SUB-Informatics-2013-6-001-016.pdf."},{"key":"2025120600251213020_j_jmc-2024-0014_ref_021","doi-asserted-by":"crossref","unstructured":"Peeva K, Kyosev Y. Fuzzy relational calculus - theory, applications and software (with CD-ROM). Advances in fuzzy systems - applications and theory. Vol. 22. Singapore: World Scientific Publishing Company; 2004.","DOI":"10.1142\/5683"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_022","doi-asserted-by":"crossref","unstructured":"De Schutter B, Heemels WPMH, Bemporad A. On the equivalence of linear complementarity problems. Operat Res Let. 2002;30(4):211\u201322.","DOI":"10.1016\/S0167-6377(02)00159-1"},{"key":"2025120600251213020_j_jmc-2024-0014_ref_023","unstructured":"Alhussaini S, Collett C, Sergeev S. Generalized Kotov\u2013Ushakov attack on tropical Stickel protocol based on modified tropical circulant matrices; 2023. https:\/\/eprint.iacr.org\/2023\/1904. Cryptology ePrint Archive, Paper 2023\/1904. https:\/\/eprint.iacr.org\/2023\/1904."}],"container-title":["Journal of Mathematical Cryptology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2024-0014\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2024-0014\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:25:41Z","timestamp":1764980741000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2024-0014\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,1]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,2,14]]},"published-print":{"date-parts":[[2024,2,14]]}},"alternative-id":["10.1515\/jmc-2024-0014"],"URL":"https:\/\/doi.org\/10.1515\/jmc-2024-0014","relation":{},"ISSN":["1862-2984"],"issn-type":[{"value":"1862-2984","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,1]]},"article-number":"20240014"}}