{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T20:50:15Z","timestamp":1792788615988,"version":"4.5.0"},"reference-count":68,"publisher":"Public Library of Science (PLoS)","issue":"9","license":[{"start":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T00:00:00Z","timestamp":1789430400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["KFU264480"],"award-info":[{"award-number":["KFU264480"]}],"id":[{"id":"https:\/\/ror.org\/00dn43547","id-type":"ROR","asserted-by":"crossref"}]}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"\n The security of modern block ciphers heavily relies on the cryptographic strength of Substitution boxes (S-boxes), which introduce non-linearity and confusion. While numerous methods for S-box generation exist, there is a continuous pursuit of designs that offer a superior and balanced profile against various cryptanalytic attacks. This paper presents a new, two-step approach to the construction of highly nonlinear S-boxes, with group-theoretic underpinnings. During the first step, we use a parameterized permutation representation of the triangle group\n \n \n \n \n \n \u0394<\/mml:mo>\n (<\/mml:mo>\n \n 2<\/mml:mtext>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n 3<\/mml:mtext>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n 4<\/mml:mtext>\n <\/mml:mrow>\n \n 3<\/mml:mtext>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n on the projective line over the finite field\n \n \n \n \n \n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n \n 2<\/mml:mtext>\n <\/mml:mrow>\n \n 5<\/mml:mtext>\n <\/mml:mrow>\n \n 7<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , denoted by\n \n \n \n \n \n \n \n \n P<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n \n 1<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msup>\n \n (<\/mml:mo>\n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n \n 2<\/mml:mtext>\n <\/mml:mrow>\n \n 5<\/mml:mtext>\n <\/mml:mrow>\n \n 7<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , to produce an initial S-box with a nonlinearity of\n \n \n \n \n \n \n 1<\/mml:mtext>\n <\/mml:mrow>\n \n 0<\/mml:mtext>\n <\/mml:mrow>\n \n 6<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . The second stage is the main innovation, in which we use a refinement process via the action of the permutation group\n \n \n \n \n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n 7406532<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:msub>\n \u00d7<\/mml:mo>\n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n \n 3<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n on this original S-box. The refinement increases the average nonlinearity to 112 and yields differential uniformity 4, linear approximation probability 0.0625, strong bit-independence behavior, and strict avalanche characteristics close to the ideal value. The proposed S-box is then incorporated into a multi-round grayscale image-encryption scheme based on fixed S-box substitution, key- and nonce-dependent position permutation, SHA-256-based keystream whitening, and a final bidirectional nonlinear diffusion layer. Experiments on Moon Surface, Airplane, and Clock images produce ciphertext entropies of 7.9968\u20137.9975, near-zero pixel correlations, NPCR values of 99.7040%\u201399.7559%, and UACI values of 33.4902%\u201333.6390%. Cropping and noise experiments further show exact recovery without attack and useful attack-dependent reconstruction under ciphertext degradation, particularly under mild cropping and low-density salt-and-pepper noise. These results demonstrate strong statistical security, differential sensitivity, and practical applicability for grayscale image protection.\n <\/jats:p>","DOI":"10.1371\/journal.pone.0357107","type":"journal-article","created":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T17:30:45Z","timestamp":1789493445000},"page":"e0357107","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["A two-stage algebraic framework for designing cryptographically strong S-boxes for secure image encryption"],"prefix":"10.1371","volume":"21","author":[{"given":"Asima","family":"Razzaque","sequence":"first","affiliation":[{"name":"King Faisal University","place":["Al-Ahsa, Saudi Arabia"],"department":["Department of Mathematics and Statistics, College of Science"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-0981-6757","authenticated-orcid":true,"given":"Hafiza Zara","family":"Mustafa","sequence":"additional","affiliation":[{"name":"University of Education","place":["Lahore, Pakistan"],"department":["Department of Mathematics, Division of Science and Technology"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dilshad","family":"Alghazzawi","sequence":"additional","affiliation":[{"name":"College of Science & Arts, King Abdul Aziz University","place":["Rabigh, Saudi Arabia"],"department":["Department of Mathematics"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ghaliah","family":"Alhamzi","sequence":"additional","affiliation":[{"name":"College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU)","place":["Riyadh, Saudi Arabia"],"department":["Department of Mathematics and Statistics"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2026,9,15]]},"reference":[{"key":"pone.0357107.ref001","first-page":"113","article-title":"Privacy protection with regard to (tele-)communications surveillance and data retention.","author":"CA Molinaro","year":"2022","journal-title":"Ius Gentium: Comparative Perspectives on Law and Justice. 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