{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T10:55:33Z","timestamp":1770461733520,"version":"3.49.0"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,3]],"date-time":"2021-08-03T00:00:00Z","timestamp":1627948800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research Foundation for the PhD","award":["XJ20BS44"],"award-info":[{"award-number":["XJ20BS44"]}]},{"name":"Scientific Research Project of Education Department of Hubei Province","award":["Q20192201"],"award-info":[{"award-number":["Q20192201"]}]},{"name":"Hubei Provincial Natural Science Foundation of China","award":["2019CFB778"],"award-info":[{"award-number":["2019CFB778"]}]},{"name":"Hubei Provincial Natural Science Foundation of China","award":["2020CFB306"],"award-info":[{"award-number":["2020CFB306"]}]},{"name":"Youth Project of Hubei School of Economics","award":["XJ201912"],"award-info":[{"award-number":["XJ201912"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Dynamic degradation occurs when chaotic systems are implemented on digital devices, which seriously threatens the security of chaos-based pseudorandom sequence generators. The chaotic degradation shows complex periodic behavior, which is often ignored by designers and seldom analyzed in theory. Not knowing the exact period of the output sequence is the key problem that affects the application of chaos-based pseudorandom sequence generators. In this paper, two cubic chaotic maps are combined, which have symmetry and reconfigurable form in the digital circuit. The dynamic behavior of the cubic chaotic map and the corresponding digital cubic chaotic map are analyzed respectively, and the reasons for the complex period and weak randomness of output sequences are studied. On this basis, the digital cubic chaotic map is optimized, and the complex periodic behavior is improved. In addition, a reconfigurable pseudorandom sequence generator based on the digital cubic chaotic map is constructed from the point of saving consumption of logical resources. Through theoretical and numerical analysis, the pseudorandom sequence generator solves the complex period and weak randomness of the cubic chaotic map after digitization and makes the output sequence have better performance and less resource consumption, which lays the foundation for applying it to the field of secure communication.<\/jats:p>","DOI":"10.3390\/sym13081420","type":"journal-article","created":{"date-parts":[[2021,8,3]],"date-time":"2021-08-03T23:37:21Z","timestamp":1628033841000},"page":"1420","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["The Dynamic Analysis of a Novel Reconfigurable Cubic Chaotic Map and Its Application in Finite Field"],"prefix":"10.3390","volume":"13","author":[{"given":"Chuanfu","family":"Wang","sequence":"first","affiliation":[{"name":"School of Information and Communication Engineering, Hubei University of Economics, Wuhan 430205, China"}]},{"given":"Yi","family":"Di","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering, Hubei University of Economics, Wuhan 430205, China"}]},{"given":"Jianyu","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering, Hubei University of Economics, Wuhan 430205, China"}]},{"given":"Jing","family":"Shuai","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering, Hubei University of Economics, Wuhan 430205, China"}]},{"given":"Yuchen","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering, Hubei University of Economics, Wuhan 430205, China"}]},{"given":"Qi","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering, Hubei University of Economics, Wuhan 430205, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic nonperiodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1080\/0161-119191865821","article-title":"Supercomputer investigations of a chaotic encryption algorithm","volume":"15","author":"Wheeler","year":"1991","journal-title":"Cryptologia"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.sigpro.2019.02.016","article-title":"A new hybrid digital chaotic system with applications in image encryption","volume":"160","author":"Alawida","year":"2019","journal-title":"Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1850084","DOI":"10.1142\/S0218127418500840","article-title":"Constructing Discrete Chaotic Systems with Positive Lyapunov Exponents","volume":"28","author":"Wang","year":"2018","journal-title":"Int. J. Bifurcat. Chaos"},{"key":"ref_5","first-page":"3900215","article-title":"A Novel Image Encryption Algorithm Based on Chaotic Sequences and Cross-Diffusion of Bits","volume":"13","author":"Shengtao","year":"2020","journal-title":"IEEE Photonics J."},{"key":"ref_6","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_7","first-page":"020504-1","article-title":"SM4 key scheme algorithm based on chaotic system","volume":"66","author":"Wang","year":"2017","journal-title":"Acta Phys. Sin."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1740","DOI":"10.1109\/TVLSI.2020.2979269","article-title":"PUF-Based Secure Chaotic Random Number Generator Design Methodology","volume":"28","author":"Kalanadhabhatta","year":"2020","journal-title":"IEEE Trans. Very Large Scale Integr. (VLSI) Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"9818520","DOI":"10.1155\/2018\/9818520","article-title":"Analysis of the time series generated by a new high-dimensional discrete chaotic system","volume":"2018","author":"Wang","year":"2018","journal-title":"Complexity"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Muhammad, A.S., and \u00d6zkaynak, F. (2021). SIEA: Secure Image Encryption Algorithm Based on Chaotic Systems Optimization Algorithms and PUFs. Symmetry, 13.","DOI":"10.3390\/sym13050824"},{"key":"ref_11","first-page":"073142","article-title":"Predicting Phase and Sensing Phase Coherence in Chaotic Systems with Machine Learning","volume":"30","author":"Zhang","year":"2020","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5942121","DOI":"10.1155\/2019\/5942121","article-title":"Constructing Digitized Chaotic Time Series with a Guaranteed Enhanced Period","volume":"2019","author":"Wang","year":"2019","journal-title":"Complexity"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1080\/0161-118991863745","article-title":"On the derivation of a \u201cChaotic\u201d encryption algorithm","volume":"8","author":"Matthews","year":"1989","journal-title":"Cryptologia"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3119","DOI":"10.1142\/S0218127405014052","article-title":"On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps","volume":"15","author":"Li","year":"2005","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_15","first-page":"49","article-title":"Period Analysis of the Logistic Map for the Finite Field","volume":"60","author":"Yang","year":"2017","journal-title":"Sci. China (Inf. Sci.)"},{"key":"ref_16","first-page":"491","article-title":"One-Dimensional Pseudo-chaotic Sequences Based on the Discrete Arnold\u2019s Cat Map Over Z3m","volume":"68","author":"Souza","year":"2021","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1817","DOI":"10.1587\/transfun.E94.A.1817","article-title":"Rounding Logistic Maps over Integers and the Properties of the Generated Sequences","volume":"94","author":"Miyazaki","year":"2011","journal-title":"IEICE Trans. Fundam. Electron. Commun. Comput. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Irfan, M., Ali, A., Khan, M.A., Ehatisham-ul-Haq, M., Mehmood Shah, S.N., Saboor, A., and Ahmad, W. (2020). Pseudorandom Number Generator (PRNG) Design Using Hyper-Chaotic Modified Robust Logistic Map (HC-MRLM). Electronics, 9.","DOI":"10.3390\/electronics9010104"},{"key":"ref_19","first-page":"1","article-title":"Dynamic Analysis of Digital Chaotic Maps via State-Mapping Networks","volume":"2019","author":"Li","year":"2019","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1109\/31.76495","article-title":"On Chaos of Digital Filters in the Real World","volume":"38","author":"Lin","year":"1991","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Alvarez, G., Amig\u00f3, J.M., Arroyo, D., and Li, S. (2011). Lessons Learnt from the Cryptanalysis of Chaos-Based Ciphers [B]. Chaos-Based Cryptography, Springer.","DOI":"10.1007\/978-3-642-20542-2_8"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1109\/TSMC.2015.2398836","article-title":"Analysis and Design of Digital Chaotic Systems with Desirable Performance via Feedback Control","volume":"45","author":"Deng","year":"2015","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1535","DOI":"10.1007\/s11071-018-4440-6","article-title":"Applications of Symbolic Dynamics in Counteracting the Dynamical Degradation of Digital Chaos","volume":"94","author":"Zheng","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1140\/epjp\/i2019-12374-y","article-title":"A novel control method to counteract the dynamical degradation of a digital chaotic sequence","volume":"134","author":"Chen","year":"2019","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s11071-019-05187-z","article-title":"Theoretical Design of Controlled Digitized Chaotic Systems with Periodic Orbit of Upper Limit Length in Digital Circuit","volume":"98","author":"Wang","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.1109\/TCSVT.2014.2369711","article-title":"Design and ARM-Embedded Implementation of A Chaotic Map-Based Real-Time Secure Video Communication System","volume":"25","author":"Lin","year":"2015","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Guyeux, C., and Bahi, J.M. (2010, January 18\u201323). Topological Chaos and Chaotic Iterations. Application to Hash functions. Proceedings of the Neural Networks (IJCNN), IEEE International Joint Conference on Neural Networks, Barcelona, Spain.","DOI":"10.1109\/IJCNN.2010.5596512"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1109\/TCSI.2016.2515398","article-title":"Theoretical Design and FPGA-Based Implementation of Higher-Dimensional Digital Chaotic Systems","volume":"63","author":"Wang","year":"2016","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2001","DOI":"10.1109\/TCYB.2014.2363168","article-title":"Cascade Chaotic System with Applications","volume":"45","author":"Zhou","year":"2015","journal-title":"IEEE Trans. Cybern."},{"key":"ref_30","first-page":"1","article-title":"A novel 2D cascade modulation couple hyperchaotic mapping for randomized image encryption","volume":"2021","author":"Shakiba","year":"2021","journal-title":"Multimed. Tools Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/0270-0255(83)90030-1","article-title":"Chaos in the cubic mapping","volume":"4","author":"Rogers","year":"1983","journal-title":"Math. Model."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1088\/0031-8949\/55\/5\/002","article-title":"Cubic map, complexity and Ljapunov exponent","volume":"55","author":"Steeb","year":"1997","journal-title":"Phys. Scr."},{"key":"ref_33","unstructured":"Klimov, A., and Shamir, A. (2002). A new class of invertible mappings. Cryptographic Hardware and Embedded Systems-CHES2002, Proceedings of the 4th International Workshop, Redwood Shores, CA, USA, 13\u201315 August 2002, Springer. Revised Papers."},{"key":"ref_34","unstructured":"Rukhin, A., Soto, J., Nechvatal, J., Smid, M., and Barker, E. (2010). A statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications."},{"key":"ref_35","unstructured":"L\u2019Ecuyer, P., and Simard, R. (2021, July 27). TestU01 A Software Library in ANSI C for Empirical Testing of Random Number Generators\u2014User\u2019s Guide, Compact Version. 16 May 2013. Available online: http:\/\/simul.iro.umontreal.ca\/testu01\/guideshorttestu01.pdf."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1420\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:39:47Z","timestamp":1760164787000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1420"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,3]]},"references-count":35,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081420"],"URL":"https:\/\/doi.org\/10.3390\/sym13081420","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,3]]}}}