{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:45:56Z","timestamp":1759970756684,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T00:00:00Z","timestamp":1738108800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>Chebyshev polynomials define a rather canonical chaotic cryptosystem and some strong attacks have been designed to that cryptosystem. We report the numerical experiments performed with multiple precision arithmetic using conventional software as gmp and mpfr to test the Chebyshev cryptosystem and Bergamo\u2019s attack. As a conclusion, we point out the relevance in the cryptosystem robustness of the number of significant digits (length) of plaintexts and the number of correct digits (precision) of the arithmetical calculations. Furthermore, after the observed experimental results, we use techniques of Numerical Analysis to explain the occurrences of the observed results and to reinforce the importance of the above-mentioned parameters \u2113 and m.<\/jats:p>","DOI":"10.3390\/cryptography9010010","type":"journal-article","created":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T07:45:12Z","timestamp":1738136712000},"page":"10","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Effectiveness of a Common Attack to Chebyshev Chaotic Encryption Scheme"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-9183-0676","authenticated-orcid":false,"given":"Xiaoqi","family":"Liu","sequence":"first","affiliation":[{"name":"Mathematics & Computer Science Program, Guangdong Technion Israel Institute of Technology, Shantou 515063, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6800-4081","authenticated-orcid":false,"given":"Guillermo","family":"Morales-Luna","sequence":"additional","affiliation":[{"name":"Computer Science Department, Cinvestav, Mexico City 07300, Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1109\/7384.963463","article-title":"Chaos-based cryptography: A brief overview","volume":"1","author":"Kocarev","year":"2001","journal-title":"IEEE Circuits Syst. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/s00034-005-2403-x","article-title":"Public-Key Encryption Based on Chebyshev Polynomials","volume":"24","author":"Kocarev","year":"2005","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Kocarev, L., and Lian, S. (2011). Chaos-Based Public-Key Cryptography. Chaos-Based Cryptography: Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-3-642-20542-2"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1136","DOI":"10.1016\/j.ins.2006.07.026","article-title":"A novel key agreement protocol based on chaotic maps","volume":"177","author":"Xiao","year":"2007","journal-title":"Inf. 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Entropy, 25.","DOI":"10.3390\/e25111516"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5110","DOI":"10.1016\/j.ins.2011.07.008","article-title":"Security analysis of the public key algorithm based on Chebyshev polynomials over the integer ring ZN","volume":"181","author":"Chen","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1382","DOI":"10.1109\/TCSI.2005.851701","article-title":"Security of public-key cryptosystems based on Chebyshev polynomials","volume":"52","author":"Bergamo","year":"2005","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_11","first-page":"2204","article-title":"Security of Public-Key Cryptosystems Based on Chebyshev Polynomials over Z\/pkZ","volume":"67","author":"Yoshioka","year":"2019","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_12","unstructured":"Free Software Foundation (2024, December 01). 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Available online: https:\/\/www.wolframalpha.com\/input\/?i=taylor+series+arccos&lk=3."}],"container-title":["Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2410-387X\/9\/1\/10\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:38:15Z","timestamp":1759919895000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2410-387X\/9\/1\/10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,29]]},"references-count":15,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["cryptography9010010"],"URL":"https:\/\/doi.org\/10.3390\/cryptography9010010","relation":{},"ISSN":["2410-387X"],"issn-type":[{"type":"electronic","value":"2410-387X"}],"subject":[],"published":{"date-parts":[[2025,1,29]]}}}