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<article article-type="Research Article">
  <front>
    <journal-meta>
      <journal-id journal-id-type="publisher">journal-of-discrete-mathematical-sciences-and-cryptography</journal-id>
      <journal-title-group>
        <journal-title>Journal of Discrete Mathematical Sciences and Cryptography</journal-title>
      </journal-title-group>
      <issn publication-format="electronic">2169-0065</issn>
      <issn publication-format="print">0972-0529</issn>
      <publisher>
        <publisher-name>Taru Publications</publisher-name>
      </publisher>
    </journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.47974/JDMSC-2698</article-id>
      <title-group>
        <article-title>A discrete mathematical framework for network intrusion detection with applications to cryptographic network security</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <name>
            <surname>Kumar</surname>
            <given-names>Ajay</given-names>
          </name>
          <aff>Department of Computer Science and Engineering, Manipal University Jaipur, Jaipur, Rajasthan, 303007, India</aff>
        </contrib>
        <contrib contrib-type="author" corresp="yes">
          <name>
            <surname>Gupta</surname>
            <given-names>Amit Kumar</given-names>
          </name>
          <aff>Department of Computer Science &amp; Engineering, Faculty of Science, Technology and Architecture (FoSTA), Manipal University Jaipur, Jaipur, Rajasthan, 303007, India</aff>
        </contrib>
        <contrib contrib-type="author">
          <name>
            <surname>Mathur</surname>
            <given-names>Priya</given-names>
          </name>
          <aff>Department of Mathematics, Poornima Institute of Engineering and Technology, Jaipur, Rajasthan, 302022, India</aff>
        </contrib>
        <contrib contrib-type="author">
          <name>
            <surname>Singh</surname>
            <given-names>Chaitanya</given-names>
          </name>
          <aff>Department of Computer Engineering, Vidhyadeep University, Surat, Gujarat, 394110, India</aff>
        </contrib>
        <contrib contrib-type="author">
          <name>
            <surname>Tiwari</surname>
            <given-names>Vipin</given-names>
          </name>
          <aff>Department of Computer Science and Engineering, Symbiosis Institute of Technology (SIT), Symbiosis International (Deemed University), Pune, Maharashtra, 412115, India</aff>
        </contrib>
      </contrib-group>
      <volume>29</volume>
      <issue>8</issue>
      <fpage>3137</fpage>
      <lpage>3147</lpage>
      <pub-date date-type="pub">
        <day>14</day>
        <month>08</month>
        <year>2026</year>
      </pub-date>
      <abstract>
        <p>As computer networks expand, robust Intrusion Detection Systems (IDS) are crucial. Traditional methods struggle with zero-day attacks and lack interpretability. This study uses the NSL-KDD dataset and three machine learning models: Random Forest, attention-based Multilayer Perceptron (MLP), and hybrid CNN–BiLSTM. Models were evaluated on accuracy, error rate, ROC–AUC, and confusion matrix. The hybrid CNN–BiLSTM achieved 77.84% accuracy; attention-based MLP reached 0.9076 AUC. SHAP (Shapley Additive Explanations) analyzed key features in the Random Forest model, enhancing transparency. Results demonstrate the framework’s improved real-time intrusion detection performance with increased explainability and practical network security applicability.</p>
      </abstract>
      <kwd-group>
        <kwd>Intrusion detection system</kwd>
        <kwd>NSL-KDD dataset</kwd>
        <kwd>CNN-BiLSTM</kwd>
        <kwd>Attention MLP</kwd>
        <kwd>Explainable AI</kwd>
        <kwd>SHAP</kwd>
        <kwd>Network security</kwd>
      </kwd-group>
      <custom-meta-group>
        <custom-meta>
          <meta-name>access</meta-name>
          <meta-value>open</meta-value>
        </custom-meta>
        <custom-meta>
          <meta-name>retracted</meta-name>
          <meta-value>no</meta-value>
        </custom-meta>
      </custom-meta-group>
    </article-meta>
  </front>
</article>
