{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T18:38:29Z","timestamp":1773513509701,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T00:00:00Z","timestamp":1757635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"UEFISCDI","award":["PN-IV-P7-7.1-PTE-2024-0505"],"award-info":[{"award-number":["PN-IV-P7-7.1-PTE-2024-0505"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>Chaotic systems, governed by deterministic nonlinear equations yet exhibiting highly complex and unpredictable behaviors, have emerged as valuable tools at the intersection of mathematics, engineering, and information security. This paper presents a comparative study of the Lorenz and R\u00f6ssler systems, focusing on their dynamic complexity and statistical independence\u2014two critical properties for applications in chaos-based cryptography. By integrating techniques from nonlinear dynamics (e.g., Lyapunov exponents, KS entropy, Kaplan\u2013Yorke dimension) and statistical testing (e.g., chi-square and Gaussian transformation-based independence tests), we provide a quantitative framework to evaluate the pseudo-randomness potential of chaotic trajectories. Our results show that the Lorenz system offers faster convergence to chaos and superior statistical independence over time, making it more suitable for rapid encryption schemes. In contrast, the R\u00f6ssler system provides complementary insights due to its simpler attractor and longer memory. These findings contribute to a multidisciplinary methodology for selecting and optimizing chaotic systems in secure communication and signal processing contexts.<\/jats:p>","DOI":"10.3390\/cryptography9030058","type":"journal-article","created":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T12:08:16Z","timestamp":1757678896000},"page":"58","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["From Chaos to Security: A Comparative Study of Lorenz and R\u00f6ssler Systems in Cryptography"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-7014-2196","authenticated-orcid":false,"given":"Alexandru","family":"Dinu","sequence":"first","affiliation":[{"name":"Faculty of Electronics, Telecommunications and Information Technology, National University of Science and Tehnology Politehnica Bucharest, 061071 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wang, L., Song, W., Di, J., Zhang, X., and Zou, C. (2025). Image Encryption Method Based on Three-Dimensional Chaotic Systems and V-Shaped Scrambling. Entropy, 27.","DOI":"10.3390\/e27010084"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Murillo-Escobar, M.A., Meranza-Castill\u00f3n, M.O., L\u00f3pez-Guti\u00e9rrez, R.M., and Cruz-Hern\u00e1ndez, C. (2019). Suggested Integral Analysis for Chaos-Based Image Cryptosystems. Entropy, 21.","DOI":"10.3390\/e21080815"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Guang, Y., Yu, L., Dong, W., Wang, Y., Zeng, J., Zhao, J., and Ding, Q. (2022). Chaos-Based Lightweight Cryptographic Algorithm Design and FPGA Implementation. Entropy, 24.","DOI":"10.3390\/e24111610"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Lawnik, M., Moysis, L., and Volos, C. (2022). Chaos-Based Cryptography: Text Encryption Using Image Algorithms. Electronics, 11.","DOI":"10.3390\/electronics11193156"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Zhang, B., and Liu, L. (2023). Chaos-Based Image Encryption: Review, Application, and Challenges. Mathematics, 11.","DOI":"10.3390\/math11112585"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1007\/11751540_127","article-title":"Revealing statistical independence of two experimental data sets: An Improvement on Spearman\u2019s Algorithm","volume":"Volume 3980","author":"Badea","year":"2006","journal-title":"Lectures Notes in Computer Science"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s12243-012-0317-8","article-title":"Unifying running-key approach and logistic map to generate enciphering sequences","volume":"68","author":"Vlad","year":"2013","journal-title":"Ann. Telecommun."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Luca, A., Vlad, A., Badea, B., and Frunzete, M. (2009, January 9\u201310). A study on statistical independence in the tent map. Proceedings of the IEEE International Symposium on Signals, Circuits and Systems (ISSCS), Ia\u0219i, Romania.","DOI":"10.1109\/ISSCS.2009.5206145"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Dinu, A., and Frunzete, M. (2023). Singularity, Observability and Statistical Independence in the Context of Chaotic Systems. Mathematics, 11.","DOI":"10.3390\/math11020305"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Dinu, A., and Frunzete, M. (2021, January 15\u201316). The Lorenz chaotic system, statistical independence and sampling frequency. Proceedings of the International Symposium on Signals, Circuits, and Systems (ISSCS), Ia\u0219i, Romania.","DOI":"10.1109\/ISSCS52333.2021.9497431"},{"key":"ref_11","first-page":"17","article-title":"The Compound Tent Map and the Connection Between Gray Codes and the Initial Condition Recovery","volume":"76","author":"Dinu","year":"2014","journal-title":"UPB Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Dinu, A., and Frunzete, M. (2025). Image Encryption Using Chaotic Maps: Development, Application, and Analysis. Mathematics, 13.","DOI":"10.3390\/math13162588"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.cosrev.2018.01.002","article-title":"Survey on hardware implementation of random number generators on FPGA: Theory and experimental analyses","volume":"27","author":"Bakiri","year":"2018","journal-title":"Comput. Sci. Rev."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ifrim, R., and Popescu, D. (2025). BPAP: FPGA Design of a RISC-like Processor for Elliptic Curve Cryptography Using Task-Level Parallel Programming in High-Level Synthesis. Cryptography, 9.","DOI":"10.3390\/cryptography9010020"},{"key":"ref_15","unstructured":"Strogatz, S. (1994). Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering (Studies in Nonlinearity), Westview Press. [1st ed.]."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Vaduva, A., Vlad, A., and Badea, B. (2016, January 9\u201310). Evaluating the performance of a test-method for statistical independence decision in the context of chaotic signals. Proceedings of the 2016 International Conference on Communications (COMM), Bucharest, Romania.","DOI":"10.1109\/ICComm.2016.7528207"},{"key":"ref_17","unstructured":"Walpole, R.E., and Myers, R.H. (1989). Probability and Statistics for Engineers and Scientists, MacMillan Publishing. [4th ed.]."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Kocarev, L., and Lian, S. (2011). Chaos-Based Cryptography: Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-3-642-20542-2"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1103\/PhysRevLett.64.821","article-title":"Synchronization in chaotic systems","volume":"64","author":"Pecora","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/S0375-9601(98)00086-3","article-title":"Cryptography with chaos","volume":"240","author":"Baptista","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2129","DOI":"10.1142\/S0218127406015970","article-title":"Some basic cryptographic requirements for chaos-based cryptosystems","volume":"16","author":"Alvarez","year":"2006","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/S0375-9601(01)00609-0","article-title":"Logistic map as a block encryption algorithm","volume":"289","author":"Kocarev","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","unstructured":"Kocarev, L., and Lian, S. (2011). Digitized chaos for pseudo-random number generation in cryptography. Chaos-Based Cryptography: Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-3-642-20542-2"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kocarev, L., and Lian, S. (2011). Formation of high-dimensional chaotic maps and their uses in cryptography. Chaos-Based Cryptography: Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-3-642-20542-2"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Kocarev, L., and Lian, S. (2011). Chaos-based hash function. Chaos-Based Cryptography: Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-3-642-20542-2"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Kocarev, L., and Lian, S. (2011). Chaos-based video encryption algorithms. Chaos-Based Cryptography: Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-3-642-20542-2"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kocarev, L., and Lian, S. (2011). Cryptanalysis of chaotic ciphers. Chaos-Based Cryptography: Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-3-642-20542-2"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1038\/nphoton.2008.227","article-title":"Fast physical random bit generation with chaotic semiconductor lasers","volume":"2","author":"Uchida","year":"2008","journal-title":"Nat. Photonics"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"20360","DOI":"10.1364\/OE.18.020360","article-title":"All-optical fast random number generator","volume":"18","author":"Li","year":"2010","journal-title":"Opt. Express"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"18763","DOI":"10.1364\/OE.18.018763","article-title":"Implementation of 140 Gb\/s true random bit generator based on a chaotic photonic integrated circuit","volume":"18","author":"Argyris","year":"2010","journal-title":"Opt. Express"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"20452","DOI":"10.1364\/OE.21.020452","article-title":"4.5 Gbps high-speed real-time physical random bit generator","volume":"21","author":"Wang","year":"2013","journal-title":"Opt. Express"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1142\/S021812749800098X","article-title":"Symmetric ciphers based on two-dimensional chaotic maps","volume":"8","author":"Fridrich","year":"1998","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1300","DOI":"10.1109\/TCSI.2006.874181","article-title":"Discrete chaos-I: Theory","volume":"53","author":"Kocarev","year":"2006","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Amig\u00f3, J.M. (2009). Chaos-based cryptography. Intelligent Computing Based on Chaos, Springer.","DOI":"10.1007\/978-3-540-95972-4_14"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Corona-Berm\u00fadez, E., Chimal-Egu\u00eda, J.C., Corona-Berm\u00fadez, U., and Rivero-\u00c1ngeles, M.E. (2023). Chaos Meets Cryptography: Developing an S-Box Design with the R\u00f6ssler Attractor. Mathematics, 11.","DOI":"10.3390\/math11224575"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Zulfiqar, N., Ahmad, T., Ghazal, T.M., Ikram, A., Gazzawe, F., Albahar, M.A., Hussein, A.H., and Khan, A.S. (2025). Securing Digital Images: A Chaos-Driven Scrambling Algorithm Using the R\u00f6ssler System. PLoS ONE, 20.","DOI":"10.1371\/journal.pone.0326202"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"11","DOI":"10.25007\/ajnu.v10n2a916","article-title":"An Efficient Method of Image Encryption Using Rossler Chaotic System","volume":"10","author":"Hamza","year":"2021","journal-title":"Acad. J. Nawroz Univ."},{"key":"ref_39","first-page":"463","article-title":"Implementation of Chaotic Rossler System in Cryptography","volume":"22","author":"Tubaki","year":"2025","journal-title":"J. Basic Sci. Eng."}],"container-title":["Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2410-387X\/9\/3\/58\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:44:31Z","timestamp":1760035471000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2410-387X\/9\/3\/58"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,12]]},"references-count":39,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["cryptography9030058"],"URL":"https:\/\/doi.org\/10.3390\/cryptography9030058","relation":{},"ISSN":["2410-387X"],"issn-type":[{"value":"2410-387X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,12]]}}}