{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,24]],"date-time":"2026-10-24T01:05:26Z","timestamp":1792803926227,"version":"4.5.0"},"reference-count":42,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T00:00:00Z","timestamp":1673568000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12161077"],"award-info":[{"award-number":["12161077"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"crossref"}]},{"name":"National Natural Science Foundation of China","award":["21JR7RE172"],"award-info":[{"award-number":["21JR7RE172"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"crossref"}]},{"name":"National Natural Science Foundation of China","award":["22JR11RE193"],"award-info":[{"award-number":["22JR11RE193"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"crossref"}]},{"name":"National Natural Science Foundation of China","award":["2021B-218"],"award-info":[{"award-number":["2021B-218"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"crossref"}]},{"name":"National Natural Science Foundation of China","award":["2021B-219"],"award-info":[{"award-number":["2021B-219"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"crossref"}]},{"name":"National Natural Science Foundation of China","award":["2021QB-109"],"award-info":[{"award-number":["2021QB-109"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"crossref"}]},{"name":"Natural Science Foundation of Gansu Province","award":["12161077"],"award-info":[{"award-number":["12161077"]}]},{"name":"Natural Science Foundation of Gansu Province","award":["21JR7RE172"],"award-info":[{"award-number":["21JR7RE172"]}]},{"name":"Natural Science Foundation of Gansu Province","award":["22JR11RE193"],"award-info":[{"award-number":["22JR11RE193"]}]},{"name":"Natural Science Foundation of Gansu Province","award":["2021B-218"],"award-info":[{"award-number":["2021B-218"]}]},{"name":"Natural Science Foundation of Gansu Province","award":["2021B-219"],"award-info":[{"award-number":["2021B-219"]}]},{"name":"Natural Science Foundation of Gansu Province","award":["2021QB-109"],"award-info":[{"award-number":["2021QB-109"]}]},{"name":"Innovation Fund Project of University in Gansu Province","award":["12161077"],"award-info":[{"award-number":["12161077"]}]},{"name":"Innovation Fund Project of University in Gansu Province","award":["21JR7RE172"],"award-info":[{"award-number":["21JR7RE172"]}]},{"name":"Innovation Fund Project of University in Gansu Province","award":["22JR11RE193"],"award-info":[{"award-number":["22JR11RE193"]}]},{"name":"Innovation Fund Project of University in Gansu Province","award":["2021B-218"],"award-info":[{"award-number":["2021B-218"]}]},{"name":"Innovation Fund Project of University in Gansu Province","award":["2021B-219"],"award-info":[{"award-number":["2021B-219"]}]},{"name":"Innovation Fund Project of University in Gansu Province","award":["2021QB-109"],"award-info":[{"award-number":["2021QB-109"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, a surjective mapping that satisfies the Li\u2013Yorke chaos in the unit area is constructed and a perturbation algorithm (disturbing its parameters and inputs through another high-dimensional chaos) is proposed to enhance the randomness of the constructed chaotic system and expand its key space. An algorithm for the composition of two systems (combining sequence based on quantum random walks with chaotic system\u2019s outputs) is designed to improve the distribution of the system outputs and a compound chaotic system is ultimately obtained. The new compound chaotic system is evaluated using some test methods such as time series complexity, autocorrelation and distribution of output frequency. The test results showed that the new system has complex dynamic behavior such as high randomicity, unpredictability and uniform output distribution. Then, a new scheme for generating pseudorandom numbers is presented utilizing the composite chaotic system. The proposed pseudorandom number generator (PRNG) is evaluated using a series test suites such as NIST sp 800-22 soft and other tools or methods. The results of tests are promising, as the proposed PRNG passed all these tests. Thus, the proposed PRNG can be used in the information security field.<\/jats:p>","DOI":"10.3390\/e25010166","type":"journal-article","created":{"date-parts":[[2023,1,16]],"date-time":"2023-01-16T02:29:55Z","timestamp":1673836195000},"page":"166","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["A Pseudorandom Number Generator Based on the Chaotic Map and Quantum Random Walks"],"prefix":"10.3390","volume":"25","author":[{"given":"Wenbo","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Electronic Information and Electrical Engineering, Tianshui Normal University, Tianshui 741000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhenhai","family":"Chang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8888-7571","authenticated-orcid":false,"given":"Caochuan","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhuozhuo","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,13]]},"reference":[{"key":"ref_1","first-page":"102509","article-title":"A new Pseudo random number generator based on generalized Newton complex map with dynamic key","volume":"53","author":"Ayubi","year":"2020","journal-title":"J. Inf. Secur. Appl."},{"key":"ref_2","first-page":"181","article-title":"S-box design method based on improved one-dimensional discrete chaotic map","volume":"2","year":"2018","journal-title":"J. Inf. Telecommun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"9971","DOI":"10.1007\/s11042-018-6584-2","article-title":"An integer wavelet transform image steganography method based on 3D sine chaotic map","volume":"78","author":"Valandar","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_4","first-page":"703","article-title":"An analogue\u2013digital mixed method for solving the dynamical degradation of digital chaotic systems","volume":"32","author":"Liu","year":"2015","journal-title":"IMA J. Math. Control Inf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/s41781-019-0034-3","article-title":"Review of High-Quality Random Number Generators","volume":"4","author":"James","year":"2020","journal-title":"Comput. Softw. Big Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1007\/s10773-018-3989-7","article-title":"Quantum Image Encryption Based on Henon Mapping","volume":"58","author":"Jiang","year":"2019","journal-title":"Int. J. Theor. Phys."},{"key":"ref_7","first-page":"2545123","article-title":"A Survey on True Random Number Generators Based on Chaos","volume":"2019","author":"Yu","year":"2019","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"105201","DOI":"10.1088\/1402-4896\/ac8b41","article-title":"A chaos-based image compression and encryption scheme using fractal coding and adaptive-thresholding sparsification","volume":"97","author":"Wang","year":"2022","journal-title":"Phys. Scr."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.cnsns.2013.06.017","article-title":"Pseudo random number generator based on quantum chaotic map","volume":"19","author":"Akhshani","year":"2014","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1007\/s10836-018-5767-0","article-title":"Security Analysis of the Efficient Chaos Pseudo-random Number Generator Applied to Video Encryption","volume":"34","author":"Ahmad","year":"2018","journal-title":"J. Electron. Test. Theory Appl. (JETTA)"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"123869","DOI":"10.1016\/j.physa.2019.123869","article-title":"Controlled alternate quantum walk-based pseudo-random number generator and its application to quantum color image encryption","volume":"547","year":"2020","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"411","DOI":"10.12988\/astp.2015.5342","article-title":"Novel secure pseudo-random number generation scheme based on two Tinkerbell maps","volume":"9","author":"Stoyanov","year":"2015","journal-title":"Adv. Stud. Theor. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1016\/j.sigpro.2010.09.014","article-title":"Optimal quantitative cryptanalysis of permutation-only multimedia ciphers against plaintext attacks","volume":"91","author":"Li","year":"2011","journal-title":"Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1323","DOI":"10.3233\/JIFS-169428","article-title":"Cryptanalysis of an image encryption algorithm based on PWLCM and inertial delayed neural network","volume":"34","author":"Ahmad","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1007\/s11128-012-0432-5","article-title":"Quantum walks: A comprehensive review","volume":"11","year":"2012","journal-title":"Quantum Inf. Process."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Aiello, L.M., Cherifi, C., Cherifi, H., Lambiotte, R., Li\u00f3, P., and Rocha, L.M. (2018, January 11\u201313). Quantum Walk Neural Networks for Graph-Structured Data. Proceedings of the Complex Networks and Their Applications VII, Cambridge, UK.","DOI":"10.1007\/978-3-030-05411-3"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"22052","DOI":"10.1038\/srep22052","article-title":"Two-walker discrete-time quantum walks on the line with percolation","volume":"6","author":"Rigovacca","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"20362","DOI":"10.1038\/srep20362","article-title":"Novel pseudo-random number generator based on quantum random walks","volume":"6","author":"Yang","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s11633-019-1173-z","article-title":"Image Encryption Application of Chaotic Sequences Incorporating Quantum Keys","volume":"17","author":"Ge","year":"2020","journal-title":"Int. J. Autom. Comput."},{"key":"ref_20","first-page":"79","article-title":"Chaos analysis of the conic in planar unit area","volume":"62","author":"Yu","year":"2013","journal-title":"Acta Phys. Sin."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1750033","DOI":"10.1142\/S021812741750033X","article-title":"Counteracting Dynamical Degradation of Digital Chaotic Chebyshev Map via Perturbation","volume":"27","author":"Liu","year":"2017","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1360\/03ys0183","article-title":"Discrete chaos in Banach spaces","volume":"48","author":"Shi","year":"2005","journal-title":"Sci. China Ser. A Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Aharonov, D., Ambainis, A., Kempe, J., and Vazirani, U. (2001, January 6\u20138). Quantum Walks On Graphs. Proceedings of the 33rd ACM Symposium on Theory of Computing, Crete, Greece.","DOI":"10.1145\/380752.380758"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_25","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_26","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1016\/j.cpc.2012.11.017","article-title":"Pseudorandom sequence generator based on the Chen chaotic system","volume":"184","author":"Hu","year":"2013","journal-title":"Comput. Phys. Commun."},{"key":"ref_27","unstructured":"(2022, December 09). Ecrypt II Yearly Report on Algorithms and Keysizes. Available online: http:\/\/www.ecrypt.eu.org\/documents\/D.SPA.20.pdf."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1016\/j.dsp.2013.01.005","article-title":"Diffusion\u2013substitution based gray image encryption scheme","volume":"23","author":"Pareek","year":"2013","journal-title":"Digit. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1209\/0295-5075\/4\/9\/004","article-title":"Recurrence Plots of Dynamical Systems","volume":"4","author":"Eckmann","year":"1987","journal-title":"Europhys. Lett. (EPL)"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"026702","DOI":"10.1103\/PhysRevE.66.026702","article-title":"Recurrence-plot-based measures of complexity and their application to heart-rate-variability data","volume":"66","author":"Marwan","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.physrep.2006.11.001","article-title":"Recurrence plots for the analysis of complex systems","volume":"438","author":"Marwan","year":"2007","journal-title":"Phys. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1152\/jappl.1994.76.2.965","article-title":"Dynamical assessment of physiological systems and states using recurrence plot strategies","volume":"76","author":"Webber","year":"1994","journal-title":"J. Appl. Physiol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/S0375-9601(03)00491-2","article-title":"Statistical complexity and disequilibrium","volume":"311","author":"Martin","year":"2003","journal-title":"Phys. Lett. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.physa.2005.05.025","article-title":"Intensive statistical complexity measure of pseudorandom number generators","volume":"356","author":"Larrondo","year":"2005","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.physa.2003.11.005","article-title":"Intensive entropic non-triviality measure","volume":"334","author":"Lamberti","year":"2004","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1016\/j.camwa.2010.05.010","article-title":"A wavelet-based tool for studying non-periodicity","volume":"60","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Bol\u00f3s, V.J., Ben\u00edtez, R., and Ferrer, R. (2020). A New Wavelet Tool to Quantify Non-Periodicity of Non-Stationary Economic Time Series. Mathematics, 8.","DOI":"10.3390\/math8050844"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.amc.2017.05.046","article-title":"The windowed scalogram difference: A novel wavelet tool for comparing time series","volume":"312","author":"Ferrer","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"100501","DOI":"10.1088\/1674-1056\/24\/10\/100501","article-title":"A perturbation method to the tent map based on Lyapunov exponent and its application","volume":"24","author":"Cao","year":"2015","journal-title":"Chin. Phys. B"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.1007\/s11071-021-06346-x","article-title":"Design of pseudo-random number generator from turbulence padded chaotic map","volume":"104","author":"Krishnamoorthi","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1080\/01611194.2018.1548390","article-title":"Designing a pseudorandom bit generator based on LFSRs and a discrete chaotic map","volume":"43","author":"Alhadawi","year":"2019","journal-title":"Cryptologia"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/166\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:05:39Z","timestamp":1760119539000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/166"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,13]]},"references-count":42,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010166"],"URL":"https:\/\/doi.org\/10.3390\/e25010166","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,13]]}}}