{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,25]],"date-time":"2026-07-25T05:06:42Z","timestamp":1784956002063,"version":"3.55.0"},"reference-count":48,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>One of the applications of dynamical systems with chaotic behavior is data encryption. Chaos-based cryptography uses chaotic dynamical systems as the basis for creating algorithms. The present article discusses a new dynamical system called M-map with its analysis: fixed points, bifurcation diagram, Lyapunov exponent, and invariant density. The obtained bifurcation diagram and the plot of the Lyapunov exponent (with a minimum value of ln2 and a maximum value of ln4) suggest that the so-called robust chaos characterizes this map. Moreover, the obtained results are compared with other dynamical systems used in cryptography. Additionally, the article proposes a new image encryption algorithm. It uses, among others, cyclically shifted S-box or saving encrypted pixels on the first or last free space in the cipher-image. The conducted analysis shows that the cipher-images are characterized by an entropy value close to 8, a correlation of adjacent pixels value close to 0, or values of Number of Pixel of Change Rate (NPCR) and Unified Average Changing Intensity (UACI) measures close to 100% and 33%, respectively.<\/jats:p>","DOI":"10.3390\/sym14050895","type":"journal-article","created":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T22:20:20Z","timestamp":1651098020000},"page":"895","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["New Chaotic System: M-Map and Its Application in Chaos-Based Cryptography"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0235-0878","authenticated-orcid":false,"given":"Marcin","family":"Lawnik","sequence":"first","affiliation":[{"name":"Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9440-7914","authenticated-orcid":false,"given":"Marek","family":"Berezowski","sequence":"additional","affiliation":[{"name":"Faculty of Chemical Engineering and Technology, Cracow University of Technology, ul. Warszawska 24, 30-155 Krak\u00f3w, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,27]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1016\/j.compeleceng.2015.11.008","article-title":"A Chaos-Based Image Encryption Algorithm with Simple Logical Functions","volume":"54","author":"Yavuz","year":"2016","journal-title":"Comput. Electr. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Safi, H.W., and Maghari, A.Y. (2017, January 16\u201317). Image Encryption Using Double Chaotic Logistic Map. Proceedings of the 2017 International Conference on Promising Electronic Technologies (ICPET), Deir El-Balah, Palestine.","DOI":"10.1109\/ICPET.2017.18"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"8124912","DOI":"10.1155\/2017\/8124912","article-title":"Cryptanalysis and Improvement of Chaos-Based Image Encryption Scheme with Circular Inter-Intra-Pixels Bit-Level Permutation","volume":"2017","author":"Fan","year":"2017","journal-title":"Math. Probl. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5190","DOI":"10.1016\/j.ijleo.2016.03.018","article-title":"Cryptanalysis of a New Image Encryption Algorithm Based on Chaos","volume":"127","year":"2016","journal-title":"Optik"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"14021","DOI":"10.1007\/s11042-016-3800-9","article-title":"Cryptanalysis of a Chaos-Based Image Encryption Scheme Combining DNA Coding and Entropy","volume":"76","author":"Su","year":"2017","journal-title":"Multimed. Tools Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5747082","DOI":"10.1155\/2020\/5747082","article-title":"On the Cryptanalysis of a Bit-Level Image Chaotic Encryption Algorithm","volume":"2020","author":"Hu","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"27483","DOI":"10.1007\/s11042-020-09163-3","article-title":"Cryptanalysis and Security Enhancement for a Chaos-Based Color Image Encryption Algorithm","volume":"79","author":"Huang","year":"2020","journal-title":"Multimed. Tools Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1080\/01611194.2020.1814447","article-title":"Cryptanalysis of Two Similar Chaos-Based Image Encryption Schemes","volume":"45","author":"Mastan","year":"2020","journal-title":"Cryptologia"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Masood, F., Ahmad, J., Shah, S.A., Jamal, S.S., and Hussain, I. (2020). A Novel Hybrid Secure Image Encryption Based on Julia Set of Fractals and 3D Lorenz Chaotic Map. Entropy, 22.","DOI":"10.3390\/e22030274"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"24801","DOI":"10.1007\/s11042-021-10695-5","article-title":"A Image Encryption Algorithm Based on Chaotic Lorenz System and Novel Primitive Polynomial S-Boxes","volume":"80","author":"Hussain","year":"2021","journal-title":"Multimed. Tools Appl."},{"key":"ref_12","unstructured":"Abraham, A., Gandhi, N., and Pant, M. (2019). An Image Encryption Method Using Henon Map and Josephus Traversal. Innovations in Bio-Inspired Computing and Applications, Springer International Publishing."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4534","DOI":"10.1109\/TCSI.2021.3082895","article-title":"Discrete Memristor Hyperchaotic Maps","volume":"68","author":"Bao","year":"2021","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.ins.2014.11.018","article-title":"2D Sine Logistic Modulation Map for Image Encryption","volume":"297","author":"Hua","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1726","DOI":"10.1109\/TII.2021.3086819","article-title":"Memristive Rulkov Neuron Model with Magnetic Induction Effects","volume":"18","author":"Li","year":"2022","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Khairullah, M., Alkahtani, A., Bin Baharuddin, M., and Al-Jubari, A. (2021). Designing 1D Chaotic Maps for Fast Chaotic Image Encryption. Electronics, 10.","DOI":"10.3390\/electronics10172116"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Lasota, A., and Mackey, M.C. (1994). Chaos, Fractals, and Noise, Springer.","DOI":"10.1007\/978-1-4612-4286-4"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Ott, E. (2002). Chaos in Dynamical Systems, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511803260"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.optlastec.2017.10.024","article-title":"A Combination Chaotic System and Application in Color Image Encryption","volume":"101","author":"Parvaz","year":"2018","journal-title":"Opt. Laser Technol."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Shahna, K., and Mohamed, A. (2018, January 11\u201313). An Image Encryption Technique Using Logistic Map and Z-Order Curve. Proceedings of the 2018 International Conference on Emerging Trends and Innovations in Engineering and Technological Research (ICETIETR), Ernakulam, India.","DOI":"10.1109\/ICETIETR.2018.8529138"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Boumaraf, M., and Merazka, F. (2018, January 25\u201326). Partial and full speech encryption schemes based on 1D chaotic maps for AMR-WB codec. Proceedings of the 2018 2nd International Conference on Natural Language and Speech Processing (ICNLSP), Algiers, Algeria.","DOI":"10.1109\/ICNLSP.2018.8374388"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Prasad, N., Ravi, V.M., and Chandrasekhar, L. (2018, January 4\u20136). Image Encryption with an Encrypted QR, Random Phase Encoding, and Logistic Map. Proceedings of the 2018 International Conference on Computer Communication and Informatics (ICCCI), Coimbatore, India.","DOI":"10.1109\/ICCCI.2018.8441219"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Jonathan Satish, T., Naga Sai Theja, M., Girish Kumar, G., and Thanikaiselvan, V. (2018, January 29\u201331). Image Encryption Using Integer Wavelet Transform, Logistic Map and XOR Encryption. Proceedings of the 2018 Second International Conference on Electronics, Communication and Aerospace Technology (ICECA), Coimbatore, India.","DOI":"10.1109\/ICECA.2018.8474930"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Singh, D.k., and Tomar, K. (2018, January 20\u201321). A Robust Color Image Encryption Algorithm in Dual Domain Using Chaotic Map. Proceedings of the 2018 Second International Conference on Inventive Communication and Computational Technologies (ICICCT), Coimbatore, India.","DOI":"10.1109\/ICICCT.2018.8473069"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Draksharam, S., Katravulapalli, D., Rohith Krishna, K., and Thanikaiselvan, V. (2018, January 29\u201331). Analysis of Hybrid Chaotic Image Encryption. Proceedings of the 2018 Second International Conference on Electronics, Communication and Aerospace Technology (ICECA), Coimbatore, India.","DOI":"10.1109\/ICECA.2018.8474807"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.optlaseng.2019.06.005","article-title":"Double-Image Encryption Based on Interference and Logistic Map under the Framework of Double Random Phase Encoding","volume":"122","author":"Liansheng","year":"2019","journal-title":"Opt. Lasers Eng."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Macovei, C., R\u0103ducanu, M., and Datcu, O. (2020, January 5\u20136). Image Encryption Algorithm Using Wavelet Packets and Multiple Chaotic Maps. Proceedings of the 2020 International Symposium on Electronics and Telecommunications (ISETC), Timisoara, Romania.","DOI":"10.1109\/ISETC50328.2020.9301088"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"52277","DOI":"10.1109\/ACCESS.2021.3069591","article-title":"Dynamic Substitution and Confusion-Diffusion-Based Noise-Resistive Image Encryption Using Multiple Chaotic Maps","volume":"9","author":"Rehman","year":"2021","journal-title":"IEEE Access"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Rama Devi, K., Janaki, V., and VijayaLaxmi, G. (2021). Encryption Methodology on Crypto Keys from Image Using Chaos Logistic Maps. Mater. Today Proc.","DOI":"10.1016\/j.matpr.2021.01.603"},{"key":"ref_30","first-page":"117","article-title":"A Secure Image Encryption Scheme Based on Cellular Automata and Chaotic Skew Tent Map","volume":"45","author":"Mondal","year":"2019","journal-title":"J. Inf. Secur. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"165661","DOI":"10.1016\/j.ijleo.2020.165661","article-title":"New Method for Fingerprint Images Encryption Using DNA Sequence and Chaotic Tent Map","volume":"224","author":"Safdarian","year":"2020","journal-title":"Optik"},{"key":"ref_32","first-page":"102711","article-title":"Chaos and Compressive Sensing Based Novel Image Encryption Scheme","volume":"58","author":"Khan","year":"2021","journal-title":"J. Inf. Secur. Appl."},{"key":"ref_33","first-page":"102626","article-title":"An Exotic Multi-Dimensional Conceptualization for Medical Image Encryption Exerting Rossler System and Sine Map","volume":"55","author":"Sangavi","year":"2020","journal-title":"J. Inf. Secur. Appl."},{"key":"ref_34","unstructured":"Ferrer, J.D. (2010, January 7\u201310). On the Inadequacy of Unimodal Maps for Cryptographic Applications. Proceedings of the RECSI 2010: IX [i.e., XI] Reuni\u00f3n Espa\u00f1ola sobre Criptolog\u00eda y Seguridad de la Informaci\u00f3n, Tarragona, Spain."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"012132","DOI":"10.1088\/1742-6596\/1141\/1\/012132","article-title":"Combined Logistic and Tent Map","volume":"1141","author":"Lawnik","year":"2018","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"012078","DOI":"10.1088\/1742-6596\/1391\/1\/012078","article-title":"Logistic Map as a Fourier\u2019s Series Expansion: Numerical Analysis","volume":"1391","author":"Lawnik","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Sahay, A., and Pradhan, C. (2017, January 6\u20138). Gauss Iterated Map Based RGB Image Encryption Approach. Proceedings of the 2017 International Conference on Communication and Signal Processing (ICCSP), Chennai, India.","DOI":"10.1109\/ICCSP.2017.8286437"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"012035","DOI":"10.1088\/1757-899X\/462\/1\/012035","article-title":"Image Compression and Encryption Using DCT and Gaussian Map","volume":"462","author":"Rahmawati","year":"2019","journal-title":"IOP Conf. Ser. Mater. Sci. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"172","DOI":"10.15388\/NA.2014.2.2","article-title":"Identification of Fast-changing Signals by Means of Adaptive Chaotic Transformations","volume":"19","author":"Berezowski","year":"2014","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"012072","DOI":"10.1088\/1742-6596\/490\/1\/012072","article-title":"The Approximation of the Normal Distribution by Means of Chaotic Expression","volume":"490","author":"Lawnik","year":"2014","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3049","DOI":"10.1103\/PhysRevLett.80.3049","article-title":"Robust Chaos","volume":"80","author":"Banerjee","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"012086","DOI":"10.1088\/1742-6596\/633\/1\/012086","article-title":"Analysis of the Chaotic Maps Generating Different Statistical Distributions","volume":"633","author":"Lawnik","year":"2015","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_43","unstructured":"Fisher, R.A., and Yates, F. (1948). Statistical Tables for Biological, Agricultural and Medical Research, Oliver & Boyd. [3rd ed.]."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Lawnik, M., and Banasik, A. (2021, January 22\u201325). Generating and Numbering Permutations with the Use of Chaotic Maps. Proceedings of the 2021 11th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS), Cracow, Poland.","DOI":"10.1109\/IDAACS53288.2021.9661044"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Lawnik, M. (2014, January 28\u201330). Generation of Numbers with the Distribution Close to Uniform with the Use of Chaotic Maps. Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications\u2014SIMULTECH, Vienna, Austria.","DOI":"10.5220\/0005090304510455"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4184","DOI":"10.1109\/ACCESS.2018.2879857","article-title":"Block-Based Optical Color Image Encryption Based on Double Random Phase Encoding","volume":"7","author":"Faragallah","year":"2019","journal-title":"IEEE Access"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"26203","DOI":"10.1007\/s11042-019-07818-4","article-title":"A Novel Chaotic Image Encryption Technique Based on Multiple Discrete Dynamical Maps","volume":"78","author":"Khan","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Wu, X., Li, Y., and Kurths, J. (2015). A New Color Image Encryption Scheme Using CML and a Fractional-Order Chaotic System. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0119660"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/895\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:01:59Z","timestamp":1760137319000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/895"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,27]]},"references-count":48,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050895"],"URL":"https:\/\/doi.org\/10.3390\/sym14050895","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,27]]}}}