{
  "abstract": "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 Δ(2,3,43) on the projective line over the finite field F257, denoted by P1(F257), to produce an initial S-box with a nonlinearity of 106. The second stage is the main innovation, in which we use a refinement process via the action of the permutation group C7406532×C3 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 7.9968–7.9975, near-zero pixel correlations, NPCR values of 99.7040%–99.7559%, and UACI values of 33.4902%–33.6390%. 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.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa, Saudi Arabia",
    "Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan",
    "Department of Mathematics, College of Science & Arts, King Abdul Aziz University, Rabigh, Saudi Arabia",
    "Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia"
  ],
  "article_type": "Research Article",
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Asima Razzaque"
    },
    {
      "affs": [
        1
      ],
      "name": "Hafiza Zara Mustafa"
    },
    {
      "affs": [
        2
      ],
      "name": "Dilshad Alghazzawi"
    },
    {
      "affs": [
        3
      ],
      "name": "Ghaliah Alhamzi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/26tvfxm4vy00k.bib",
  "cited_by": 0,
  "doi": "10.1371/journal.pone.0357107",
  "fulltext_url": null,
  "id": "26tvfxm4vy00k",
  "issns": [
    "1932-6203"
  ],
  "landing_url": "https://doi.org/10.1371/journal.pone.0357107",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "1702|Cryptographic Implementations and Security"
    ],
    [
      "plos",
      "/Computer and information sciences/Cryptography"
    ],
    [
      "plos",
      "/Computer and information sciences/Cryptography/Encryption"
    ],
    [
      "plos",
      "/Computer and information sciences/Data management/Data visualization/Infographics/Graphs"
    ],
    [
      "plos",
      "/Computer and information sciences/Digital imaging/Grayscale"
    ],
    [
      "plos",
      "/Physical sciences/Mathematics/Algebra/Algebraic structures"
    ],
    [
      "plos",
      "/Physical sciences/Mathematics/Cryptography"
    ],
    [
      "plos",
      "/Physical sciences/Mathematics/Cryptography/Encryption"
    ],
    [
      "plos",
      "/Physical sciences/Mathematics/Discrete mathematics/Combinatorics/Permutation"
    ],
    [
      "plos",
      "/Physical sciences/Mathematics/Statistics/Statistical noise/Gaussian noise"
    ],
    [
      "plos",
      "/Physical sciences/Physics/Thermodynamics/Entropy"
    ]
  ],
  "openalex": "W7213335731",
  "pdf_url": "https://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0357107&type=printable",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": "Public Library of Science",
  "ris": "https://research.draconitic.com/paper/26tvfxm4vy00k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:49Z",
      "native_id": "10.1371/journal.pone.0357107",
      "source": "plos",
      "url": "https://doi.org/10.1371/journal.pone.0357107"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "plos:algebra",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "plos:combinatorics",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "plos:statistics",
      "subfield": "statistics"
    }
  ],
  "title": "A two-stage algebraic framework for designing cryptographically strong S-boxes for secure image encryption",
  "title_text": "A two-stage algebraic framework for designing cryptographically strong S-boxes for secure image encryption",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:17:49Z",
  "url": "https://research.draconitic.com/paper/26tvfxm4vy00k",
  "venue": "PLOS One",
  "versions": []
}