{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,10]],"date-time":"2024-08-10T02:17:45Z","timestamp":1723256265504},"reference-count":64,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,16]],"date-time":"2020-04-16T00:00:00Z","timestamp":1586995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010221","name":"HEC","doi-asserted-by":"publisher","award":["NRPU-7433"],"id":[{"id":"10.13039\/501100010221","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"We propose an image encryption scheme based on quasi-resonant Rossby\/drift wave triads (related to elliptic surfaces) and Mordell elliptic curves (MECs). By defining a total order on quasi-resonant triads, at a first stage we construct quasi-resonant triads using auxiliary parameters of elliptic surfaces in order to generate pseudo-random numbers. At a second stage, we employ an MEC to construct a dynamic substitution box (S-box) for the plain image. The generated pseudo-random numbers and S-box are used to provide diffusion and confusion, respectively, in the tested image. We test the proposed scheme against well-known attacks by encrypting all gray images taken from the USC-SIPI image database. Our experimental results indicate the high security of the newly developed scheme. Finally, via extensive comparisons we show that the new scheme outperforms other popular schemes.<\/jats:p>","DOI":"10.3390\/e22040454","type":"journal-article","created":{"date-parts":[[2020,4,21]],"date-time":"2020-04-21T07:23:06Z","timestamp":1587453786000},"page":"454","source":"Crossref","is-referenced-by-count":17,"title":["Image Encryption Using Elliptic Curves and Rossby\/Drift Wave Triads"],"prefix":"10.3390","volume":"22","author":[{"given":"Ikram","family":"Ullah","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}]},{"given":"Umar","family":"Hayat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}]},{"ORCID":"http:\/\/orcid.org\/0000-0003-3360-0304","authenticated-orcid":false,"given":"Miguel D.","family":"Bustamante","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"105818","DOI":"10.1016\/j.optlastec.2019.105818","article-title":"Evolutionary-Based Image Encryption using RNA Codons Truth Table","volume":"121","author":"Mahmud","year":"2020","journal-title":"Optics Laser Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s11071-014-1445-7","article-title":"An Efficient Chaotic Image Encryption Based on Alternate Circular S-boxes","volume":"78","author":"Zhang","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.aeue.2012.07.004","article-title":"A Hybrid Chaotic System and Cyclic Elliptic Curve for Image Encryption","volume":"67","author":"Niu","year":"2013","journal-title":"AEU-Int. J. Electron. Commun."},{"key":"ref_4","first-page":"1","article-title":"Novel Image Encryption Based on Quantum Walks","volume":"5","author":"Yang","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Zhong, H., Chen, X., and Tian, Q. (2019). An Improved Reversible Image Transformation using K-Means Clustering and Block Patching. Information, 10.","DOI":"10.3390\/info10010017"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1109\/MMUL.2017.3051512","article-title":"Cryptanalyzing an Image-Scrambling Encryption Algorithm of Pixel Bits","volume":"24","author":"Li","year":"2017","journal-title":"IEEE MultiMedia"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.sigpro.2017.10.004","article-title":"Medical Image Encryption using High-Speed Scrambling and Pixel Adaptive Diffusion","volume":"144","author":"Hua","year":"2018","journal-title":"Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.sigpro.2016.10.002","article-title":"On the Cryptanalysis of Fridrich\u2019s Chaotic Image Encryption Scheme","volume":"132","author":"Xie","year":"2017","journal-title":"Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5790189","DOI":"10.1155\/2017\/5790189","article-title":"A Novel Fuzzy Encryption Technique Based on Multiple Right Translated AES Gray S-Boxes and Phase Embedding","volume":"2017","author":"Azam","year":"2017","journal-title":"Secur. Commun. Netw."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"77740","DOI":"10.1109\/ACCESS.2018.2884013","article-title":"A Double-Image Encryption Scheme Based on Amplitude-Phase Encoding and Discrete Complex Random Transformation","volume":"6","author":"Luo","year":"2018","journal-title":"IEEE Access"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"10374","DOI":"10.1038\/srep10374","article-title":"Compressive Optical Image Encryption","volume":"5","author":"Li","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"8660","DOI":"10.1109\/ACCESS.2018.2890116","article-title":"Image Encryption using Josephus Problem and Filtering Diffusion","volume":"7","author":"Hua","year":"2019","journal-title":"IEEE Access"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1080\/09500340.2019.1571249","article-title":"Image Encryption using the Random FrDCT and the Chaos-Based Game of Life","volume":"66","author":"Wu","year":"2019","journal-title":"J. Modern Opt."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"39781","DOI":"10.1109\/ACCESS.2020.2975880","article-title":"Comparison of Pre and Post-Action of a Finite Abelian Group Over Certain Nonlinear Schemes","volume":"8","author":"Yousaf","year":"2020","journal-title":"IEEE Access"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107280","DOI":"10.1016\/j.sigpro.2019.107280","article-title":"A Novel Image Encryption System Merging Fractional-Order Edge Detection and Generalized Chaotic Maps","volume":"167","author":"Ismail","year":"2020","journal-title":"Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"8694678","DOI":"10.1155\/2019\/8694678","article-title":"Image Encryption with Double Spiral Scans and Chaotic Maps","volume":"2019","author":"Tang","year":"2019","journal-title":"Secur. Commun. Netw."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3875","DOI":"10.1109\/ACCESS.2019.2958336","article-title":"Secure Image Transmission using Chaotic-Enhanced Elliptic Curve Cryptography","volume":"8","author":"Abdelfatah","year":"2019","journal-title":"IEEE Access"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Yu, J., Guo, S., Song, X., Xie, Y., and Wang, E. (2020). Image Parallel Encryption Technology Based on Sequence Generator and Chaotic Measurement Matrix. Entropy, 22.","DOI":"10.3390\/e22010076"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"67095","DOI":"10.1109\/ACCESS.2018.2874336","article-title":"A Novel Image Compression-Encryption Scheme Based on Chaos and Compression Sensing","volume":"6","author":"Zhu","year":"2018","journal-title":"IEEE Access"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"ElKamchouchi, D.H., Mohamed, H.G., and Moussa, K.H. (2020). A Bijective Image Encryption System Based on Hybrid Chaotic Map Diffusion and DNA Confusion. Entropy, 22.","DOI":"10.3390\/e22020180"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3039","DOI":"10.1016\/j.sigpro.2013.04.021","article-title":"Image Encryption using a New Parametric Switching Chaotic System","volume":"93","author":"Zhou","year":"2013","journal-title":"Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.ins.2018.12.048","article-title":"Cosine-Transform-Based Chaotic System for Image Encryption","volume":"480","author":"Hua","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3735","DOI":"10.1016\/j.cnsns.2014.02.029","article-title":"Image Encryption Based on Synchronization of Fractional Chaotic Systems","volume":"19","author":"Xu","year":"2014","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"852","DOI":"10.1016\/j.procs.2015.07.494","article-title":"An Enhanced Image Encryption Algorithm using Fractional Chaotic Systems","volume":"57","author":"Ahmad","year":"2015","journal-title":"Procedia Comput. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2121","DOI":"10.1007\/s11071-014-1798-y","article-title":"A Fast Image Encryption Algorithm Based on Chaotic Map and Lookup Table","volume":"79","author":"Cheng","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.sigpro.2016.03.021","article-title":"A Novel Image Encryption Scheme Based on Substitution-Permutation Network and Chaos","volume":"128","author":"Belazi","year":"2016","journal-title":"Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/s13319-016-0084-9","article-title":"A New Image Encryption Scheme Based on Dynamic S-boxes and Chaotic Maps","volume":"7","author":"Rehman","year":"2016","journal-title":"3D Res."},{"key":"ref_28","first-page":"637","article-title":"A New Method of Encryption Algorithm Based on Chaos and ECC","volume":"7","author":"Jia","year":"2016","journal-title":"J. Inf. Hiding Multimedia Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.sigpro.2017.06.010","article-title":"An Image Encryption Scheme Based on Elliptic Curve Pseudo Random and Advanced Encryption System","volume":"141","author":"Toughi","year":"2017","journal-title":"Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.optlastec.2013.07.009","article-title":"Cryptanalyzing an Image Encryption Scheme Based on Hybrid Chaotic System and Cyclic Elliptic Curve","volume":"56","author":"Liu","year":"2014","journal-title":"Opt. Laser Technol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/j.sigpro.2018.10.011","article-title":"A Novel Image Encryption Scheme Based on an Elliptic Curve","volume":"155","author":"Hayat","year":"2019","journal-title":"Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2402","DOI":"10.1016\/j.cnsns.2012.12.024","article-title":"Complete Classification of Discrete Resonant Rossby\/Drift Wave Triads on Periodic Domains","volume":"18","author":"Bustamante","year":"2013","journal-title":"Commun. Nonlinear Sci. Numer Simul."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"104896","DOI":"10.1016\/j.cnsns.2019.104896","article-title":"Discrete Resonant Rossby\/Drift Wave Triads: Explicit Parameterisations and a Fast Direct Numerical Search Algorithm","volume":"79","author":"Hayat","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3421725","DOI":"10.1155\/2018\/3421725","article-title":"An Injective S-Box Design Scheme over an Ordered Isomorphic Elliptic Curve and Its Characterization","volume":"2018","author":"Azam","year":"2018","journal-title":"Secur. Commun. Netw."},{"key":"ref_35","first-page":"3","article-title":"On the scale of atmospheric motions","volume":"17","author":"Charney","year":"1948","journal-title":"Geophys. Public"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1063\/1.862083","article-title":"Pseudo-three-dimensional turbulence in magnetized nonuniform plasma","volume":"21","author":"Hasegawa","year":"1978","journal-title":"Phys. Fluids"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1017\/S0022112010000510","article-title":"Modulational instability of Rossby and drift waves and generation of zonal jets","volume":"654","author":"Connaughton","year":"2010","journal-title":"J. Fluid Mech."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"083011","DOI":"10.1088\/1367-2630\/15\/8\/083011","article-title":"Percolation Transition in the Kinematics of Nonlinear Resonance Broadening in Charney\u2013Hasegawa\u2013Mima Model of Rossby Wave Turbulence","volume":"15","author":"Harris","year":"2013","journal-title":"New J. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Galperin, B., and Read, P.L. (2019). Zonal Jets: Phenomenology, Genesis, and Physics, Cambridge University Press.","DOI":"10.1017\/9781107358225"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1137\/16M1077593","article-title":"The Arithmetic Geometry of Resonant Rossby Wave Triads","volume":"1","author":"Kopp","year":"2017","journal-title":"SIAM J. Appl. Algebra Geomet."},{"key":"ref_41","unstructured":"Washington, L.C. (2003). Elliptic Curves Number Theory and Cryptography, Discrete Mathematics and Its Applications, Chapman and Hall\/CRC, University of Maryland College Park. [2nd ed.]."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.1631\/FITEE.1800434","article-title":"Efficient Construction of S-boxes Based on a Mordell Elliptic Curve Over a Finite Field","volume":"20","author":"Azam","year":"2019","journal-title":"Front. Inf. Technol. Electron. Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/BF00203967","article-title":"The Structured Design of Cryptographically Good S-boxes","volume":"3","author":"Adams","year":"1990","journal-title":"J. Cryptol."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Matsui, M. (1994). Linear cryptanalysis method of DES cipher. Advances in Cryptology, Proceedings of the Workshop on the Theory and Application of of Cryptographic Techniques (EURO-CRYPT-93), Lofthus, Norway, 23\u201327 May 1993, Springer.","DOI":"10.1007\/3-540-48285-7_33"},{"key":"ref_45","unstructured":"Webster, A., and Tavares, S.E. (1985). On the design of S-boxes. Conference on the Theory and Application of Cryptographic Techniques, Springer."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/BF00630563","article-title":"Differential Cryptanalysis of DES-like Cryptosystems","volume":"4","author":"Biham","year":"1991","journal-title":"J. Cryptol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2115","DOI":"10.1007\/s11071-018-4478-5","article-title":"Chaotic S-box: Six-Dimensional Fractional Lorenz\u2013Duffing Chaotic System and O-shaped Path Scrambling","volume":"94","author":"Ye","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1007\/s11760-016-1007-1","article-title":"A New S-box Construction Method Based on the Fractional-Order Chaotic Chen System","volume":"11","year":"2017","journal-title":"Signal Image Video Process."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1007\/s11071-016-3099-0","article-title":"A Novel Approach for Strong S-Box Generation Algorithm Design Based on Chaotic Scaled Zhongtang System","volume":"87","author":"Zengin","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1438","DOI":"10.1016\/j.ijleo.2016.11.152","article-title":"A Simple yet Efficient S-box Method Based on Chaotic Sine Map","volume":"130","author":"Belazi","year":"2017","journal-title":"Optik"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3317","DOI":"10.1007\/s00521-017-3287-y","article-title":"Construction of robust substitution boxes based on chaotic systems","volume":"31","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Liu, L., Zhang, Y., and Wang, X. (2018). A Novel Method for Constructing the S-box Based on Spatiotemporal Chaotic Dynamics. Appl. Sci., 8.","DOI":"10.3390\/app8122650"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1007\/s11277-018-5698-1","article-title":"A Method of Generating 8 \u00d7 8 Substitution Boxes Based on Elliptic Curves","volume":"101","author":"Hayat","year":"2018","journal-title":"Wirel. Pers. Commun."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Wang, X., \u00c7avu\u015fo\u011flu, \u00dc., Kacar, S., Akgul, A., Pham, V.T., Jafari, S., Alsaadi, F.E., and Nguyen, X.Q. (2019). S-box Based Image Encryption Application using a Chaotic System without Equilibrium. Appl. Sci., 9.","DOI":"10.3390\/app9040781"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"9389065","DOI":"10.1155\/2018\/9389065","article-title":"Sine-Cosine Optimization-Based Bijective Substitution-Boxes Construction using Enhanced Dynamics of Chaotic Map","volume":"2018","author":"Alzaidi","year":"2018","journal-title":"Complexity"},{"key":"ref_56","unstructured":"(2020, February 21). USC-SIPI Image Database. Available online: http:\/\/sipi.usc.edu\/database\/database.php."},{"key":"ref_57","unstructured":"(2020, April 15). Available online: https:\/\/github.com\/ikram702314\/Results."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1950263","DOI":"10.1142\/S0217984919502634","article-title":"Chaotic Image Encryption Algorithm Based on Pseudo-Random Bit Sequence and DNA Plane","volume":"33","author":"Wang","year":"2019","journal-title":"Modern Phys. Lett. B"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.sigpro.2017.04.006","article-title":"Color Image Encryption Based on Chaotic Systems and Elliptic Curve ElGamal Scheme","volume":"141","author":"Wu","year":"2017","journal-title":"Signal Process."},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Wan, Y., Gu, S., and Du, B. (2020). A New Image Encryption Algorithm Based on Composite Chaos and Hyperchaos Combined with DNA Coding. Entropy, 22.","DOI":"10.3390\/e22020171"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"2333","DOI":"10.1007\/s11071-016-2648-x","article-title":"A Joint Color Image Encryption and Compression Scheme Based on Hyper-Chaotic System","volume":"84","author":"Tong","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.cnsns.2013.06.031","article-title":"An Image Encryption Scheme Based on Rotation Matrix Bit-Level Permutation and Block Diffusion","volume":"19","author":"Zhang","year":"2014","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1142\/S0219498803000532","article-title":"A Polynomial Description of the Rijndael Advanced Encryption Standard","volume":"2","author":"Rosenthal","year":"2003","journal-title":"J. Algebra. Its Appl."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"23","DOI":"10.15388\/Informatica.2009.235","article-title":"Key-Dependent S-box Generation in AES Block Cipher system","volume":"20","author":"Kazlauskas","year":"2009","journal-title":"Informatica"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/454\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,26]],"date-time":"2024-06-26T23:14:20Z","timestamp":1719443660000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/454"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,16]]},"references-count":64,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["e22040454"],"URL":"https:\/\/doi.org\/10.3390\/e22040454","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,16]]}}}