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We also offer countermeasures against recent attacks by Panny and by Brown and Monico.<\/jats:p>","DOI":"10.1515\/jmc-2024-0005","type":"journal-article","created":{"date-parts":[[2024,9,28]],"date-time":"2024-09-28T02:21:06Z","timestamp":1727490066000},"source":"Crossref","is-referenced-by-count":6,"title":["Tropical cryptography III: Digital signatures"],"prefix":"10.1515","volume":"18","author":[{"given":"Jiale","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, The City College of New York , NY 10031 , New York , United States"}]},{"given":"Dima","family":"Grigoriev","sequence":"additional","affiliation":[{"name":"CNRS, Math\u00e9matiques, Universit\u00e9 de Lille , 59655 , Villeneuve d\u2019Ascq , France"}]},{"given":"Vladimir","family":"Shpilrain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The City College of New York , NY 10031 , New York , United States"}]}],"member":"374","published-online":{"date-parts":[[2024,9,27]]},"reference":[{"key":"2025120600251204340_j_jmc-2024-0005_ref_001","doi-asserted-by":"crossref","unstructured":"Grigoriev D, Shpilrain V. Tropical cryptography. Comm Algebra. 2014;42:2624\u201332.","DOI":"10.1080\/00927872.2013.766827"},{"key":"2025120600251204340_j_jmc-2024-0005_ref_002","doi-asserted-by":"crossref","unstructured":"Grigoriev D, Shpilrain V. Tropical cryptography II: extensions by homomorphisms. Comm Algebra. 2019;47:4224\u20139.","DOI":"10.1080\/00927872.2019.1581213"},{"key":"2025120600251204340_j_jmc-2024-0005_ref_003","doi-asserted-by":"crossref","unstructured":"Habeeb M, Kahrobaei D, Koupparis C, Shpilrain V. Public key exchange using semidirect product of (semi)groups. In: ACNS 2013, Lecture Notes in Computer Science. Vol. 7954. 2013. p. 475\u201386.","DOI":"10.1007\/978-3-642-38980-1_30"},{"key":"2025120600251204340_j_jmc-2024-0005_ref_004","doi-asserted-by":"crossref","unstructured":"Kahrobaei D, Shpilrain V. Using semidirect product of (semi)groups in public key cryptography. In: CiE 2016, Lecture Notes in Computer Science. Vol. 9709. 2016. p. 132\u201341.","DOI":"10.1007\/978-3-319-40189-8_14"},{"key":"2025120600251204340_j_jmc-2024-0005_ref_005","doi-asserted-by":"crossref","unstructured":"Kotov M, Ushakov A. Analysis of a key exchange protocol based on tropical matrix algebra. J Math Cryptology. 2018;12:137\u201341.","DOI":"10.1515\/jmc-2016-0064"},{"key":"2025120600251204340_j_jmc-2024-0005_ref_006","unstructured":"Kim KH, Roush FW. Factorization of polynomials in one variable over the tropical semiring. https:\/\/arxiv.org\/pdf\/math\/0501167.pdf."},{"key":"2025120600251204340_j_jmc-2024-0005_ref_007","doi-asserted-by":"crossref","unstructured":"Chen J, Grigoriev D, Shpilrain V. Tropical cryptography III: digital signatures. https:\/\/eprint.iacr.org\/archive\/versions\/2023\/1475.","DOI":"10.1515\/jmc-2024-0005"},{"key":"2025120600251204340_j_jmc-2024-0005_ref_008","unstructured":"Panny L. Forging tropical signatures. https:\/\/eprint.iacr.org\/2023\/1748\/."},{"key":"2025120600251204340_j_jmc-2024-0005_ref_009","unstructured":"Brown DRL, Monico C. More forging (and patching) of tropical signatures. https:\/\/eprint.iacr.org\/2023\/1837."},{"key":"2025120600251204340_j_jmc-2024-0005_ref_010","unstructured":"G\u00e9raud-Stewart R, Naccache D, Yifrach-Stav O. Fiat-Shamir goes tropical. https:\/\/eprint.iacr.org\/2023\/1954."},{"key":"2025120600251204340_j_jmc-2024-0005_ref_011","doi-asserted-by":"crossref","unstructured":"Fiat A, Shamir A. How to prove yourself: Practical solutions to identification and signature problems. In: Proceedings on Advances in Cryptology - CRYPTO\u201986. Springer; 1987. p. 186\u201394.","DOI":"10.1007\/3-540-47721-7_12"},{"key":"2025120600251204340_j_jmc-2024-0005_ref_012","doi-asserted-by":"crossref","unstructured":"Butkovic P. Max-linear systems: theory and algorithms. London: Springer-Verlag; 2010.","DOI":"10.1007\/978-1-84996-299-5"},{"key":"2025120600251204340_j_jmc-2024-0005_ref_013","doi-asserted-by":"crossref","unstructured":"Theobald T. On the frontiers of polynomial computations in tropical geometry. 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