{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T16:45:47Z","timestamp":1765039547603,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,31]],"date-time":"2020-12-31T00:00:00Z","timestamp":1609372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Image encryption is an excellent method for the protection of image content. Most authors used the permutation-substitution model to encrypt\/decrypt the image. Chaos-based image encryption methods are used in this model to shuffle the rows\/columns and change the pixel values. In parallel, authors proposed permutation using non-chaotic methods and have displayed good results in comparison to chaos-based methods. In the current article, a new image encryption algorithm is designed using combination of Newton-Raphson\u2019s method (non-chaotic) and general Bischi-Naimzadah duopoly system as a hyperchaotic two-dimensional map. The plain image is first shuffled by using Newton-Raphson\u2019s method. Next, a secret matrix with the same size of the plain image is created using general Bischi-Naimzadah duopoly system. Finally, the XOR between the secret matrix and the shuffled image is calculated and then the cipher image is obtained. Several security experiments are executed to measure the efficiency of the proposed algorithm, such as key space analysis, correlation coefficients analysis, histogram analysis, entropy analysis, differential attacks analysis, key sensitivity analysis, robustness analysis, chosen plaintext attack analysis, computational analysis, and NIST statistical Tests. Compared to many recent algorithms, the proposed algorithm has good security efficiency.<\/jats:p>","DOI":"10.3390\/e23010057","type":"journal-article","created":{"date-parts":[[2020,12,31]],"date-time":"2020-12-31T10:10:37Z","timestamp":1609409437000},"page":"57","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Cryptographic Algorithm Using Newton-Raphson Method and General Bischi-Naimzadah Duopoly System"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9819-9860","authenticated-orcid":false,"given":"Abdelrahman","family":"Karawia","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6663","DOI":"10.1007\/s11227-019-02878-7","article-title":"An image encryption method based on chaos system and AES algorithm","volume":"75","author":"Arab","year":"2019","journal-title":"J. Supercomput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1850047","DOI":"10.1142\/S0218127418500475","article-title":"A new chaotic image encryption scheme using breadth-first search and dynamic diffusion","volume":"28","author":"Yin","year":"2018","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1049\/iet-ipr.2016.0906","article-title":"Cryptographic algorithm based on pixel shuffling and dynamical chaotic economic map","volume":"12","author":"Askar","year":"2018","journal-title":"IET Image Process"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Karawia, A. (2018). Encryption Algorithm of Multiple-Image Using Mixed Image Elements and Two Dimensional Chaotic Economic Map. Entropy, 20.","DOI":"10.3390\/e20100801"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Askar, S., Karawia, A., Al-Khedhairi, A., and Alammar, F. (2019). An Algorithm of Image Encryption Using Logistic and Two-Dimensional Chaotic Economic Maps. Entropy, 1.","DOI":"10.3390\/e21010044"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2086","DOI":"10.1049\/iet-ipr.2018.5142","article-title":"Image encryption based on Fisher-Yates shuffling and three dimensional chaotic economic map","volume":"13","author":"Karawia","year":"2019","journal-title":"IET Image Process"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1007\/s11071-017-3698-4","article-title":"Lossless chaotic color image cryptosystem based on DNA encryption and entropy","volume":"90","author":"Wu","year":"2017","journal-title":"Nonlinear Dynam."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.ins.2016.02.041","article-title":"A novel lossless color image encryption scheme using 2d dwt and 6d hyperchaotic system","volume":"349","author":"Wu","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ivanov, G., Nikolov, N., and Nikova, S. (2015, January 3\u20134). Cryptographically strong S-boxes generated by modified immune algorithm. Proceedings of the International Conference on Cryptography and Information Security in the Balkans, Koper, Slovenia.","DOI":"10.1007\/978-3-319-29172-7_3"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.1631\/FITEE.1800434","article-title":"Efficient construction of a substitution box based on a Mordell elliptic curve over a finite field","volume":"20","author":"Azam","year":"2019","journal-title":"Front. Inform. Technol. El"},{"key":"ref_11","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 Multimed. Signal Process."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"2445","DOI":"10.1016\/j.ijleo.2015.06.018","article-title":"An image encryption scheme based on a new hyperchaotic finance system","volume":"126","author":"Tonga","year":"2015","journal-title":"Optik"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"55540","DOI":"10.1109\/ACCESS.2020.2981771","article-title":"Quadratic function chaotic system and its application on digital image encryption","volume":"8","author":"Guo","year":"2020","journal-title":"IEEE Access"},{"key":"ref_15","first-page":"3124","article-title":"Implementation and testing of high-speed cmos true random number generators based on chaotic systems","volume":"57","author":"Pareschi","year":"2010","journal-title":"IEEE Trans. Circuits-I"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1007\/s11071-015-2008-2","article-title":"A novel color image encryption algorithm based on spatial permutation and quantum chaotic map","volume":"81","author":"Seyedzadeh","year":"2015","journal-title":"Nonlinear Dynam."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2622","DOI":"10.1109\/TCYB.2015.2483621","article-title":"N-dimensional discrete cat map generation using laplace expansions","volume":"46","author":"Wu","year":"2016","journal-title":"IEEE Trans. Cybern."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1016\/j.physa.2005.01.001","article-title":"Security analysis of a chaos-based image encryption algorithm","volume":"351","author":"Lian","year":"2005","journal-title":"Phys. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.physleta.2006.10.081","article-title":"Cryptanalysis of chaotic stream cipher","volume":"363","author":"Skrobek","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1016\/S0375-9601(98)00425-3","article-title":"Cryptanalyzing chaotic secure communications using return maps","volume":"245","author":"Yang","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_21","unstructured":"Shakiba, A. (2019). A randomized CPA-secure asymmetric-key chaotic color image encryption scheme based on the Chebyshev mappings and one-time pad. J. King Saud Univ. Comput. Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Xiao, S., Yu, Z., and Deng, Y. (2020). Design and analysis of a novel chaos-based image encryption algorithm via switch control mechanism. Secur. Commun. Netw., 2020.","DOI":"10.1155\/2020\/7913061"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/j.sigpro.2017.08.020","article-title":"A novel bit-level image encryption algorithm based on 2d-LICM hyperchaotic map","volume":"143","author":"Cao","year":"2018","journal-title":"Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"34773","DOI":"10.1007\/s11042-019-08071-5","article-title":"A novel randomized one-dimensional chaotic chebyshev mapping for chosen plaintext attack secure image encryption with a novel chaotic breadth first traversal","volume":"78","author":"Shakiba","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.sigpro.2017.03.011","article-title":"A new color image encryption using combination of the 1d chaotic map","volume":"138","author":"Pak","year":"2017","journal-title":"Signal Process."},{"key":"ref_26","first-page":"25173","article-title":"A secure image cryptosystem using 2D arnold cat map and logistic map","volume":"8","author":"Rajendran","year":"2016","journal-title":"Int. J. Pharm. Technol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4700","DOI":"10.1016\/j.ijleo.2014.05.023","article-title":"A fast image encryption method by using chaotic 3D cat maps","volume":"125","author":"Gu","year":"2014","journal-title":"Optik"},{"key":"ref_28","first-page":"208","article-title":"Improving Image Encryption Using 3D Cat Map and Turing Machine","volume":"7","author":"Mohamed","year":"2016","journal-title":"Int. J. Adv. Comput. Sci. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"100023","DOI":"10.1016\/j.csfx.2020.100023","article-title":"A Self-perturbed Pseudo-random Sequence Generator Based on Hyperchaos","volume":"4","author":"Zhao","year":"2019","journal-title":"Chaos Soliton Fract. X"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3518","DOI":"10.1142\/S0218127411029641","article-title":"Breaking an image encryption algorithm based on chaos","volume":"21","author":"Li","year":"2011","journal-title":"Int. J. Bifurcat. Chaos"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.1016\/j.sigpro.2011.10.023","article-title":"A novel colour image encryption algorithm based on chaos","volume":"92","author":"Wang","year":"2012","journal-title":"Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1319","DOI":"10.1007\/s11071-018-4426-4","article-title":"A novel plaintext-related image encryption scheme using hyper-chaotic system","volume":"94","author":"Li","year":"2018","journal-title":"Nonlinear Dynam."},{"key":"ref_33","unstructured":"Lindell, Y., and Katz, J. (2014). Introduction to Modern Cryptography, CRC Press."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2383","DOI":"10.1007\/s11071-012-0626-5","article-title":"Breaking a novel colour image encryption algorithm based on chaos","volume":"70","author":"Li","year":"2012","journal-title":"Nonlinear Dynam."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Ding, L., and Ding, Q. (2020). A Novel Image Encryption Scheme Based on 2D Fractional Chaotic Map, DWT and 4D Hyper-chaos. Electronics, 9.","DOI":"10.3390\/electronics9081280"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Enzeng, D., Zengqiang, C., Zhuzhi, Y., and Zaiping, C. (2008, January 19\u201321). A chaotic images encryption algorithm with the key mixing proportion factor. Proceedings of the 2008 International Conference on Information Management, Innovation Management and Industrial Engineering, Taipei, Taiwan.","DOI":"10.1109\/ICIII.2008.25"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Hosny, K. (2020). Multimedia Security Using Chaotic Maps: Principles and Methodologies. Studies in Computational Intelligence, Springer.","DOI":"10.1007\/978-3-030-38700-6"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Askar, S., and Al-Khedhairi, A. (2020). Local and Global Dynamics of a Constraint Profit Maximization for Bischi-Naimzada Competition Duopoly Game. Mathematics, 8.","DOI":"10.3390\/math8091458"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2077","DOI":"10.1016\/j.jss.2012.04.002","article-title":"Cryptanalyzing a chaos-based image encryption algorithm using alternate structure","volume":"85","author":"Zhang","year":"2012","journal-title":"J. Syst. Softw."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Shakiba, A. (2020). A novel randomized bit-level two-dimensional hyperchaotic image encryption algorithm. Multimed. Tools Appl.","DOI":"10.1007\/s11042-020-09434-z"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Escobar, M., Castillon, M., Gutierrez, R., and Hernandez, C. (2019). Suggested Integral Analysis for Chaos-Based Image Cryptosystems. Entropy, 21.","DOI":"10.3390\/e21080815"},{"key":"ref_42","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_43","doi-asserted-by":"crossref","unstructured":"Rukhin, A., Soto, J., Nechvatal, J., Smid, M., and Barker, E. (2001, May 15). A statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, Available online: http:\/\/www.nist.gov\/manuscript-publication-search.cfm?pub_id=151222.","DOI":"10.6028\/NIST.SP.800-22"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/57\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:48:27Z","timestamp":1760179707000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/57"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,31]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["e23010057"],"URL":"https:\/\/doi.org\/10.3390\/e23010057","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,12,31]]}}}