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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Plant Sci.</journal-id>
<journal-title>Frontiers in Plant Science</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Plant Sci.</abbrev-journal-title>
<issn pub-type="epub">1664-462X</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.3389/fpls.2025.1634408</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Plant Science</subject>
<subj-group>
<subject>Original Research</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>A privacy-protecting eggplant disease detection framework based on the YOLOv11n-12D model</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname>Han</surname>
<given-names>Jiao</given-names>
</name>
<role content-type="https://credit.niso.org/contributor-roles/writing-original-draft/"/>
<role content-type="https://credit.niso.org/contributor-roles/funding-acquisition/"/>
<role content-type="https://credit.niso.org/contributor-roles/methodology/"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Wu</surname>
<given-names>Zhenzhen</given-names>
</name>
<role content-type="https://credit.niso.org/contributor-roles/writing-original-draft/"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Ding</surname>
<given-names>Yandong</given-names>
</name>
<uri xlink:href="https://loop.frontiersin.org/people/3074901/overview"/>
<role content-type="https://credit.niso.org/contributor-roles/writing-review-editing/"/>
<role content-type="https://credit.niso.org/contributor-roles/conceptualization/"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Guo</surname>
<given-names>Yantong</given-names>
</name>
<role content-type="https://credit.niso.org/contributor-roles/data-curation/"/>
<role content-type="https://credit.niso.org/contributor-roles/writing-original-draft/"/>
</contrib>
<contrib contrib-type="author" corresp="yes">
<name>
<surname>Fu</surname>
<given-names>Rui</given-names>
</name>
<xref ref-type="author-notes" rid="fn001">
<sup>*</sup>
</xref>
<uri xlink:href="https://loop.frontiersin.org/people/3056321/overview"/>
<role content-type="https://credit.niso.org/contributor-roles/investigation/"/>
<role content-type="https://credit.niso.org/contributor-roles/writing-review-editing/"/>
<role content-type="https://credit.niso.org/contributor-roles/supervision/"/>
</contrib>
</contrib-group>
<aff id="aff1">
<institution>Weifang University of Science and Technology</institution>, <addr-line>Weifang</addr-line>,&#xa0;<country>China</country>
</aff>
<author-notes>
<fn fn-type="edited-by">
<p>Edited by: <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/2863105/overview">Hao Wang</ext-link>, Laoshan National Laboratory, China</p>
</fn>
<fn fn-type="edited-by">
<p>Reviewed by: <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/2945309/overview">Maragatharajan M</ext-link>, VIT Bhopal University, India</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/2969566/overview">Matthew Oladipupo</ext-link>, University of Salford, United Kingdom</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/3176037/overview">Xiaotong Li</ext-link>, China University of Petroleum (East China), China</p>
</fn>
<fn fn-type="corresp" id="fn001">
<p>*Correspondence: Rui Fu, <email xlink:href="mailto:furui19891209@wfust.edu.cn">furui19891209@wfust.edu.cn</email>
</p>
</fn>
</author-notes>
<pub-date pub-type="epub">
<day>10</day>
<month>10</month>
<year>2025</year>
</pub-date>
<pub-date pub-type="collection">
<year>2025</year>
</pub-date>
<volume>16</volume>
<elocation-id>1634408</elocation-id>
<history>
<date date-type="received">
<day>27</day>
<month>05</month>
<year>2025</year>
</date>
<date date-type="accepted">
<day>03</day>
<month>09</month>
<year>2025</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2025 Han, Wu, Ding, Guo and Fu.</copyright-statement>
<copyright-year>2025</copyright-year>
<copyright-holder>Han, Wu, Ding, Guo and Fu</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/">
<p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p>
</license>
</permissions>
<abstract>
<p>The growing global population and rising concerns about food security highlight the critical need for intelligent agriculture. Among various technologies, plant disease detection is vital but faces challenges in balancing data privacy and model accuracy. To address this, we propose a novel privacy-preserving eggplant disease detection system with high accuracy. First, we introduce a lightweight 3D chaotic cube-based image encryption method that ensures security with low computational cost. Second, a streamlined YOLOv11n-12D framework is employed to optimize detection performance on resource-constrained devices. Finally, the encryption and detection modules are integrated into a real-time, secure, and accurate identification system.Experimental results show our framework achieves near-ideal encryption security (entropy=7.6195, Number of Pixel Change Rate(NPCR)=99.63%, Unified Average Changing Intensity(UACI)=32.85%) with 23&#xd7; faster encryption (0.0127s) versus existing methods. The distilled YOLOv11n-12D model maintains teacher-level accuracy (mAP@0.5=0.849) at 3.6&#xd7; the speed of YOLOv12s (2.7ms/inference), with +6.5% mAP improvement for small disease detection (e.g., thrips). This system balances privacy and real-time performance for smart agriculture applications.</p>
</abstract>
<kwd-group>
<kwd>image encryption</kwd>
<kwd>eggplant disease detection</kwd>
<kwd>YOLOv11n-12D</kwd>
<kwd>privacy protection</kwd>
<kwd>intelligent agriculture</kwd>
</kwd-group>
<counts>
<fig-count count="10"/>
<table-count count="4"/>
<equation-count count="45"/>
<ref-count count="56"/>
<page-count count="16"/>
<word-count count="7828"/>
</counts>
<custom-meta-wrap>
<custom-meta>
<meta-name>section-in-acceptance</meta-name>
<meta-value>Sustainable and Intelligent Phytoprotection</meta-value>
</custom-meta>
</custom-meta-wrap>
</article-meta>
</front>
<body>
<sec id="s1" sec-type="intro">
<label>1</label>
<title>Introduction</title>
<p>With the rapid advancement of agricultural digitalization, crop disease detection has become critical for ensuring food security and improving agricultural productivity (<xref ref-type="bibr" rid="B11">Elijah et&#xa0;al., 2018</xref>; <xref ref-type="bibr" rid="B6">Cornia, 1985</xref>). In many remote or underdeveloped regions, due to the lack of professional expertise and detection equipment, farmers often transmit crop images to external agricultural centers for manual or automated analysis (<xref ref-type="bibr" rid="B4">Baldi and La Porta, 2020</xref>). However, existing methods frequently struggle to balance model accuracy with data privacy protection. In addition, the detection systems must operate efficiently on resource-constrained devices typical of rural environments, while ensuring secure handling of sensitive data during transmission and storage. Consequently, the development of efficient, automated, and privacy-preserving disease detection systems that are both lightweight and reliable is crucial for promoting smart agriculture.</p>
<p>In recent years, deep learning-based object detection algorithms have achieved remarkable progress in plant disease recognition (<xref ref-type="bibr" rid="B44">Senthil Pandi et&#xa0;al., 2024</xref>; <xref ref-type="bibr" rid="B41">Rav&#xec; et&#xa0;al., 2017</xref>; <xref ref-type="bibr" rid="B53">Zhang et&#xa0;al., 2017</xref>). Lightweight models, particularly the YOLO series, have attracted considerable attention for their fast detection speeds and high accuracy (<xref ref-type="bibr" rid="B54">Zhang et&#xa0;al., 2024</xref>; <xref ref-type="bibr" rid="B14">He et&#xa0;al., 2024</xref>; <xref ref-type="bibr" rid="B33">Liu et&#xa0;al., 2022</xref>). <xref ref-type="bibr" rid="B43">Sangaiah et&#xa0;al. (2024)</xref> proposed T-YOLO-Rice, based on YOLOv4, to improve small-target detection such as rice leaf spots, outperforming YOLOv7 but remaining limited to a single task. To address diverse diseases and class imbalance, <xref ref-type="bibr" rid="B42">Roy and Bhaduri (2022)</xref> developed Dense-YOLOv4 by integrating DenseNet and an enhanced PANet, achieving 96.20% mAP and 93.61% F1-score for mango disease detection, and demonstrating generalization to grape and tomato diseases. <xref ref-type="bibr" rid="B29">Lin et&#xa0;al. (2023)</xref> built YOLO-Tobacco based on YOLOX-Tiny by adding HMU and CBAM modules, improving outdoor tobacco leaf detection (80.56% AP, 69 FPS), although its adaptability to multiple diseases remains limited. Building upon these advances, <xref ref-type="bibr" rid="B25">Li et&#xa0;al. (2023a)</xref> introduced MG-YOLO, integrating multi-head self-attention, BiFPN, and GhostCSP modules, achieving 98.3% accuracy at 0.009 seconds per image and surpassing YOLOv5 by 6.8% in complex environments. In addition to single-task detection, recent studies have explored joint detection and tracking paradigms (<xref ref-type="bibr" rid="B26">Li et&#xa0;al., 2023b</xref>, <xref ref-type="bibr" rid="B27">2024</xref>), leveraging reinforcement learning to achieve object recognition and continuous tracking in dynamic environments. For example, <xref ref-type="bibr" rid="B28">Li et&#xa0;al. (2025)</xref> proposed a reinforcement learning-based joint detection and tracking paradigm for compact HFSWR target detection and tracking, which effectively improves detection probability and tracking performance.</p>
<p>In image tasks related to object detection, image enhancement has also emerged as an important research direction in recent years. For example, researchers have proposed a reinforcement learning-based human visual perception-driven image enhancement method (<xref ref-type="bibr" rid="B34">Luo, 2024</xref>). <xref ref-type="bibr" rid="B30">Liu et&#xa0;al. (2025a)</xref> introduced a framework that cascades an aerial image enhancement module with AC3Net, while <xref ref-type="bibr" rid="B49">Xiao et&#xa0;al. (2024)</xref> proposed a neuromorphic computing-based underwater image enhancement network (UIEN), which simulates visual system perception and employs unsupervised learning to address multiple types of underwater image degradation and validate its effectiveness. Despite these significant advances in image enhancement, most existing studies still overlook data privacy issues, as unencrypted images transmitted over networks are vulnerable to theft or misuse. This further highlights the necessity of integrating image encryption with recognition.</p>
<p>Therefore, with increasing emphasis on data privacy, researchers have begun integrating image encryption with disease detection to achieve end-to-end security without compromising performance. <xref ref-type="bibr" rid="B39">Qin et&#xa0;al. (2014)</xref> proposed SecSIFT, a method that performs SIFT feature extraction directly within the encrypted domain, effectively safeguarding sensitive image data while maintaining high detection accuracy and computational efficiency. Building on this idea, <xref ref-type="bibr" rid="B35">Man et&#xa0;al. (2021)</xref> integrated convolutional neural networks with chaotic encryption, enabling intelligent privacy protection for both image and text data, and laying the foundation for secure image processing in agriculture. <xref ref-type="bibr" rid="B24">Kumar et&#xa0;al. (2021)</xref> introduced SP2F, a privacy-preserving framework combining blockchain and deep learning, with a two-level privacy engine and stacked LSTM networks to improve UAV data authentication and resilience. Furthermore, <xref ref-type="bibr" rid="B21">Kethineni and Gera (2023)</xref> proposed an IoT security model that integrates sparse capsule autoencoders and attention-based GRUs for lightweight detection and data protection, achieving 99.9% accuracy and F1 score, highlighting its potential for agricultural data security.</p>
<p>Our work aims to develop a lightweight deep learning model for precise crop disease detection and robust image-level privacy protection. Optimized for resource-constrained edge devices, it ensures real-time, high-precision identification of various disease types. Additionally, to secure data transmission, we integrate a novel image encryption scheme based on a 3D chaotic cube, effectively preventing unauthorized access without compromising detection performance. Our model has been comprehensively evaluated on real-world datasets and outperforms existing methods in detection accuracy, computational overhead, and privacy protection. This solution offers a practical and secure pathway for smart agriculture applications. Our approach addresses two key challenges in plant disease detection: data privacy and detection accuracy.</p>
<p>Our main contributions are as follows:</p>
<list list-type="bullet">
<list-item>
<p>We propose an encryption model combining SHA-256, a 3D Logistic Map, pixel permutation, and XOR operations, ensuring both strong security and high efficiency. Compared to traditional Advanced Encryption Standard(AES) and Rivest-Shamir-Adleman(RSA), our method offers a larger key space, enhanced attack resistance, and millisecond-level encryption speeds, making it well-suited for edge and mobile devices in agriculture. Security evaluations using entropy, Structural Similarity Index Measure(SSIM), Number of Pixel Change Rate(NPCR), and Unified Average Changing Intensity(UACI) confirm its balanced performance.</p>
</list-item>
<list-item>
<p>We present a knowledge distillation framework with YOLOv12s as the teacher and YOLOv11n as the student. The distilled student model, YOLOv11n-12D, inherits enhanced detection capabilities while maintaining a lightweight structure. To address class imbalance and improve small lesion detection, Focal Loss and CIoU Loss are incorporated during training. Experimental results show that YOLOv11n-12D outperforms existing lightweight models in precision, recall, F1 score, and mAP, while achieving real-time inference speed.</p>
</list-item>
<list-item>
<p>We develop an end-to-end system in which farmers encrypt images locally, transmit them wirelessly to a diagnostic center, and receive encrypted detection results. This framework ensures data security and scalability across various crop scenarios, effectively integrating deep learning and encryption technologies. The overall architecture is shown in <xref ref-type="fig" rid="f1">
<bold>Figure&#xa0;1</bold>
</xref>.</p>
</list-item>
</list>
<fig id="f1" position="float">
<label>Figure&#xa0;1</label>
<caption>
<p>Overview of the process.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g001.tif">
<alt-text content-type="machine-generated">Flowchart illustrating an image transmission process including three main parts: Encryption, Decryption, and Detection. The Encryption Part involves key preparation, image preprocessing, 3D key generation, permutation, and XOR encryption, followed by saving the encrypted image. The Decryption Module loads the key, reconstructs a 3D cube, performs channel-wise decryption, and reconstructs a 2D image. The image is then stored in a database. The Detection Section involves the YOLOv11n-12D detection module, teacher-student distillation training, and encrypting detection results to produce a diagnosis report at the receiver side.</alt-text>
</graphic>
</fig>
<p>The rest of this paper is organized as follows: Section 2 reviews related work. Section 3 details the encryption method. Section 4 introduces the detection model. Section 5 describes data processing and optimization. Section 6 presents experiments and analysis. Section 7 concludes the paper.Section 8 highlights the system&#x2019;s significance, practical value, and limitations.</p>
</sec>
<sec id="s2">
<label>2</label>
<title>Related work</title>
<p>Traditional encryption algorithms such as AES and DES are inadequate for real-time protection of agricultural images due to high dimensionality, redundancy, and computational overhead associated with such data. Although deep learning has achieved success in plant disease detection tasks, most existing studies overlook privacy concerns during image transmission and processing. To contextualize the proposed integrated system, this section reviews key image encryption techniques and plant disease detection approaches.</p>
<sec id="s2_1">
<label>2.1</label>
<title>Data encryption techniques</title>
<p>In response to the need for secure image transmission in agriculture, several encryption techniques have been developed, each aiming to balance security and efficiency. Key representative methods are summarized below. <xref ref-type="bibr" rid="B36">Niyat et&#xa0;al. (2017)</xref> proposed an encryption scheme based on non-uniform cellular automata (CA) and hyper-chaotic mapping, enhancing key space and attack resistance. <xref ref-type="bibr" rid="B23">Kulalvaimozhi et&#xa0;al. (2020)</xref> introduced a method combining homomorphic encryption (NHE) and enhanced discrete wavelet transform (EDWT), improving both security and compression efficiency. <xref ref-type="bibr" rid="B38">Priyanka et&#xa0;al. (2024)</xref> employed 3D chaotic mapping and Huffman coding for medical image encryption. <xref ref-type="bibr" rid="B45">Sha et&#xa0;al. (2024)</xref> developed an IoT-oriented image encryption scheme utilizing graph data structures and logic gate mechanisms to strengthen attack protection. <xref ref-type="bibr" rid="B10">Ding et&#xa0;al. (2022)</xref> proposed a GAN-based key generation model, significantly improving key security.</p>
<p>
<xref ref-type="bibr" rid="B9">Devi et&#xa0;al. (2024)</xref> proposed a DWT-SVD watermarking and PSMD symmetric encryption scheme to enhance UAV image security. While effective, its reliance on symmetric keys may pose challenges in key management and attack resistance. <xref ref-type="bibr" rid="B56">Zhou et&#xa0;al. (2024)</xref> applied compressed sensing and a two-dimensional hyperchaotic coupled Fourier oscillator system (2D-HCFOS) to improve encryption speed and security, achieving promising simulation results. Chen et&#xa0;al. <xref ref-type="bibr" rid="B55">Zhou et&#xa0;al. (2025)</xref> introduced a 2D super-attractor Logistic coupled chaotic model (2D-SALC), outperforming existing methods in chaos and security metrics. However, further validation, including integration with YOLO models and assessment of encryption impact on detection accuracy, remains needed.</p>
</sec>
<sec id="s2_2">
<label>2.2</label>
<title>Deep learning-based disease monitoring</title>
<p>Deep learning has shown strong results in plant disease detection <xref ref-type="bibr" rid="B3">Attri et&#xa0;al. (2023)</xref>, with notable performance across crops like rice, wheat, tomato, and grape. <xref ref-type="bibr" rid="B18">Jia et&#xa0;al. (2023)</xref> improved YOLOv7 for rice pest detection by integrating MobileNetV3 and coordinate attention, achieving 92.3% accuracy and 93.7% mAP@0.5. However, its performance in complex backgrounds still faces challenges. To address this, <xref ref-type="bibr" rid="B8">Deng et&#xa0;al. (2023)</xref> enhanced YOLOv5s and YOLOv7-tiny models for better accuracy and speed, enabling mobile deployment. <xref ref-type="bibr" rid="B31">Liu and Wang (2020)</xref> optimized YOLOv3 with an image pyramid for better multi-scale detection in tomato disease recognition. <xref ref-type="bibr" rid="B52">Zhang et&#xa0;al. (2023)</xref> proposed RYWD and SSA networks for wheat Fusarium head blight detection, improving accuracy and precision by 11.8% and 10.7%. <xref ref-type="bibr" rid="B48">Wu et&#xa0;al. (2023)</xref> combined YOLOv5 with HRNet for grape stem localization, achieving 92% accuracy in bunch detection and 90.2% in stem recognition. While these methods show improvements, their performance under complex field conditions still requires further refinement.</p>
<p>For eggplant disease detection, <xref ref-type="bibr" rid="B32">Liu et&#xa0;al. (2025b)</xref> enhanced YOLOv8n with the YOLO-RDM model, improving accuracy and robustness. <xref ref-type="bibr" rid="B16">Huang et&#xa0;al. (2024)</xref> proposed YOLOv8-E, which enhanced detection accuracy and small target recognition while reducing computational complexity. MR et&#xa0;al. <xref ref-type="bibr" rid="B13">Haque and Sohel (2022)</xref> used a dual-stream architecture combining CNN-SVM and CNN-Softmax, outperforming traditional models. Despite these advances, challenges remain in achieving high accuracy, robustness, and data security.</p>
<p>Despite significant progress in image encryption and plant disease detection, several critical gaps remain. Most existing studies treat encryption and detection as separate processes, lacking a unified solution that simultaneously ensures privacy protection and detection accuracy. Moreover, few works consider the resource constraints of real-time processing on edge devices. Many YOLO-based methods either overlook the impact of encryption on feature extraction or employ models that are too heavy for mobile deployment. Therefore, there is a need for a unified lightweight framework that guarantees image security while enabling efficient disease detection. To address this gap, we propose an integrated system that combines 3D chaotic cube encryption with the YOLOv11n-12D detection model, aiming to enhance both detection performance and data security.</p>
</sec>
</sec>
<sec id="s3">
<label>3</label>
<title>3D chaotic cube encryption scheme</title>
<p>In eggplant disease detection, image encryption is essential for data security by preventing unauthorized access, tampering, and maintaining integrity. Ciphertext transmission enhances system security and reduces the risk of cyberattacks. This section presents a novel image encryption method based on a 3D Chaotic Cube Encryption Scheme, which consists of four steps: preparation of the key and image, generation of 3D key and index array, permutation encryption and XOR operation, and save the encrypted image and key. Compared to frequency- and chaos-based methods (<xref ref-type="bibr" rid="B19">Jui-Cheng and Guo, 2000</xref>; <xref ref-type="bibr" rid="B2">Armand Eyebe Fouda et&#xa0;al., 2014</xref>; <xref ref-type="bibr" rid="B17">Jammula et&#xa0;al., 2022</xref>), the proposed scheme offers stronger resistance to attacks and superior performance. <xref ref-type="fig" rid="f2">
<bold>Figure&#xa0;2</bold>
</xref> illustrates the encryption framework, and <xref ref-type="fig" rid="f3">
<bold>Figure&#xa0;3</bold>
</xref> shows the original and encrypted eggplant images.</p>
<fig id="f2" position="float">
<label>Figure&#xa0;2</label>
<caption>
<p>Encryption flowchart.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g002.tif">
<alt-text content-type="machine-generated">Flowchart of an encryption process with four main steps. Step one is preparation of key and image, involving channel separation and replication. Step two is generation of a 3D key and index array through a chaotic map and random sequence. Step three is permutation-based encryption with row and column transformations and rotations. Step four is generation of ciphertext cube via XOR operation.</alt-text>
</graphic>
</fig>
<fig id="f3" position="float">
<label>Figure&#xa0;3</label>
<caption>
<p>
<bold>(A-D)</bold> show the original images, <bold>(E-H)</bold> display the encrypted images.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g003.tif">
<alt-text content-type="machine-generated">The image shows a grid of eight 3D visualizations labeled as &#x201c;Stacked Triple Cubes,&#x201d; arranged in two rows of four. Panels A to D feature colorful, textured cubes, each with varied surface patterns resembling natural scenes. Panels E to H depict similar cubes in grayscale, lacking the detailed textures of the top row. Each cube is set within a 3D coordinate grid.</alt-text>
</graphic>
</fig>
<sec id="s3_1">
<label>3.1</label>
<title>Preparation of the key and image</title>
<p>Prior to encryption, a 64-bit hexadecimal key and the target image are provided. The key is then processed using a hash function to generate the initial values for the chaotic system. The provided 64-bit hexadecimal string key hex is first converted into binary, and its SHA-256 hash value is computed: H<sub>K</sub>=SHA-256(key_hex). Assume the original image <italic>img</italic> has a size of 128 &#xd7; 128. Extract the R, G, and B channels separately as R(<italic>i,j</italic>), G(<italic>i,j</italic>), and B(<italic>i,j</italic>), where <italic>i,j</italic> &#x2208; {0,127}. The result is as shown in <xref ref-type="disp-formula" rid="eq1">Equations 1</xref>&#x2013;<xref ref-type="disp-formula" rid="eq3">3</xref>. Then expand the image into a 128 &#xd7; 128 &#xd7; 128 3D cube and <italic>K</italic> &#x2208; {0,127}:</p>
<disp-formula id="eq1">
<label>(1)</label>
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<mml:mrow>
<mml:msub>
<mml:mi mathvariant="normal">C</mml:mi>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
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<mml:mi>k</mml:mi>
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<mml:mo>=</mml:mo>
<mml:mi mathvariant="normal">R</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
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<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq2">
<label>(2)</label>
<mml:math display="block" id="M2">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="normal">C</mml:mi>
<mml:mi>G</mml:mi>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>k</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mi mathvariant="normal">G</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq3">
<label>(3)</label>
<mml:math display="block" id="M3">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="normal">C</mml:mi>
<mml:mi>B</mml:mi>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>k</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mi mathvariant="normal">B</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<p>The three channel cubes are concatenated into a one-dimensional bitstream. The image hash is then computed as shown in <xref ref-type="disp-formula" rid="eq4">Equation 4</xref>:</p>
<disp-formula id="eq4">
<label>(4)</label>
<mml:math display="block" id="M4">
<mml:mrow>
<mml:msub>
<mml:mi>H</mml:mi>
<mml:mi>I</mml:mi>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mtext>SHA&#x2212;</mml:mtext>
<mml:mn>256</mml:mn>
<mml:mo stretchy="false">(</mml:mo>
<mml:mtext>img</mml:mtext>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<p>To initialize the chaotic system, extract the first, middle, and last 64 bits from the bitstream, convert them into decimal values <italic>x</italic>
<sub>0</sub>
<italic>,y</italic>
<sub>0</sub>
<italic>,z</italic>
<sub>0</sub>, and normalize each to the range (0,1). These values are used as initial conditions for the chaotic system, x<sub>0</sub>,y<sub>0</sub>,z<sub>0</sub> as defined in <xref ref-type="disp-formula" rid="eq5">Equations 5</xref>&#x2013;<xref ref-type="disp-formula" rid="eq7">7</xref>:</p>
<disp-formula id="eq5">
<label>(5)</label>
<mml:math display="block" id="M5">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mtext>int</mml:mtext>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>H</mml:mi>
<mml:mi>K</mml:mi>
</mml:msub>
<mml:mo stretchy="false">[</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>:</mml:mo>
<mml:mn>16</mml:mn>
<mml:mo stretchy="false">]</mml:mo>
<mml:mo>,</mml:mo>
<mml:mn>16</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mn>2</mml:mn>
<mml:mrow>
<mml:mn>64</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq6">
<label>(6)</label>
<mml:math display="block" id="M6">
<mml:mrow>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mtext>int</mml:mtext>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>H</mml:mi>
<mml:mi>K</mml:mi>
</mml:msub>
<mml:mo stretchy="false">[</mml:mo>
<mml:mn>16</mml:mn>
<mml:mo>:</mml:mo>
<mml:mn>32</mml:mn>
<mml:mo stretchy="false">]</mml:mo>
<mml:mo>,</mml:mo>
<mml:mn>16</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mn>2</mml:mn>
<mml:mrow>
<mml:mn>64</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq7">
<label>(7)</label>
<mml:math display="block" id="M7">
<mml:mrow>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mtext>int</mml:mtext>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>H</mml:mi>
<mml:mi>K</mml:mi>
</mml:msub>
<mml:mo stretchy="false">[</mml:mo>
<mml:mn>48</mml:mn>
<mml:mo>:</mml:mo>
<mml:mn>64</mml:mn>
<mml:mo stretchy="false">]</mml:mo>
<mml:mo>,</mml:mo>
<mml:mn>16</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mn>2</mml:mn>
<mml:mrow>
<mml:mn>64</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>
</sec>
<sec id="s3_2">
<label>3.2</label>
<title>Generation of 3D key and index array</title>
<list list-type="bullet">
<list-item>
<p>To generate chaotic sequences, the 3D Logistic Map is employed with the following initial conditions, as defined in <xref ref-type="disp-formula" rid="eq8">Equations 8</xref>&#x2013;<xref ref-type="disp-formula" rid="eq10">10</xref>:</p>
</list-item>
</list>
<disp-formula id="eq8">
<label>(8)</label>
<mml:math display="block" id="M8">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mtext>r</mml:mtext>
<mml:mi>x</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mi>&#x3b2;</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq9">
<label>(9)</label>
<mml:math display="block" id="M9">
<mml:mrow>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mtext>r</mml:mtext>
<mml:mi>y</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mi>&#x3b3;</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq10">
<label>(10)</label>
<mml:math display="block" id="M10">
<mml:mrow>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mtext>r</mml:mtext>
<mml:mi>z</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</disp-formula>
<p>Then use <italic>x</italic>
<sub>1</sub>
<italic>, y</italic>
<sub>1</sub>
<italic>, z</italic>
<sub>1</sub> in the equations again to calculate the next values of <italic>x</italic>
<sub>2</sub>
<italic>,y</italic>
<sub>2</sub>
<italic>,z</italic>
<sub>2</sub> repeat this process to generate a chaotic sequence array as shown in <xref ref-type="disp-formula" rid="eq11">Equations 11</xref>&#x2013;<xref ref-type="disp-formula" rid="eq13">13</xref>:</p>
<disp-formula id="eq11">
<label>(11)</label>
<mml:math display="block" id="M11">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>+</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mtext>r</mml:mtext>
<mml:mi>x</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mi>&#x3b2;</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq12">
<label>(12)</label>
<mml:math display="block" id="M12">
<mml:mrow>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>+</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mtext>r</mml:mtext>
<mml:mi>y</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mi>&#x3b3;</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq13">
<label>(13)</label>
<mml:math display="block" id="M13">
<mml:mrow>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>+</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mtext>r</mml:mtext>
<mml:mi>z</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</disp-formula>
<p>Here, <italic>r<sub>x</sub>,r<sub>y</sub>,r<sub>z</sub>
</italic> &#x2208; (3, 57, 4), Take the values from the chaotic interval. These <italic>&#x3b1;, &#x3b2;, &#x3b3;</italic> control the coupling degree of the system. The 3D Logistic Map is iterated one million times, and the initial steps are discarded to eliminate transient effects. This process generates three long chaotic sequences. <xref ref-type="fig" rid="f4">
<bold>Figure&#xa0;4</bold>
</xref> shows the resulting random sequences, chaotic sequences as defined in <xref ref-type="disp-formula" rid="eq14">Equations 14</xref>&#x2013;<xref ref-type="disp-formula" rid="eq16">16</xref>:</p>
<fig id="f4" position="float">
<label>Figure&#xa0;4</label>
<caption>
<p>Random sequence numbers.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g004.tif">
<alt-text content-type="machine-generated">Three line graphs show the first 200 points of X, Y, and Z sequences. The X-sequence is red, the Y-sequence is green, and the Z-sequence is blue. Each graph has a similar range from 0 to 1 on the y-axis, indicating fluctuating data points. The x-axis ranges from 0 to 200.</alt-text>
</graphic>
</fig>
<disp-formula id="eq14">
<label>(14)</label>
<mml:math display="block" id="M14">
<mml:mrow>
<mml:mstyle mathsize="normal">
<mml:mi>X</mml:mi>
</mml:mstyle>
<mml:mo>=</mml:mo>
<mml:mo>{</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#x2026;</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>N</mml:mi>
</mml:msub>
<mml:mo>}</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq15">
<label>(15)</label>
<mml:math display="block" id="M15">
<mml:mrow>
<mml:mstyle mathsize="normal">
<mml:mi>Y</mml:mi>
</mml:mstyle>
<mml:mo>=</mml:mo>
<mml:mo>{</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#x2026;</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mi>N</mml:mi>
</mml:msub>
<mml:mo>}</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<disp-formula id="eq16">
<label>(16)</label>
<mml:math display="block" id="M16">
<mml:mrow>
<mml:mstyle mathsize="normal">
<mml:mi>Z</mml:mi>
</mml:mstyle>
<mml:mo>=</mml:mo>
<mml:mo>{</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#x2026;</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mi>N</mml:mi>
</mml:msub>
<mml:mo>}</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<p>Map the values to the range [0, 255] to form the 3D key: <italic>K</italic>(<italic>x,y,z</italic>)= [<italic>X</italic>(<italic>x,y,z</italic>) &#xd7; 256].</p>
<list list-type="bullet">
<list-item>
<p>To construct the index arrays, X is sorted to obtain A = argsort(X), and Y is sorted to obtain B = argsort(Y), Z is normalized to the range [0, 3], which is used for rotation: C = [Z &#xd7; 4].</p>
</list-item>
</list>
</sec>
<sec id="s3_3">
<label>3.3</label>
<title>Permutation encryption and XOR operation</title>
<list list-type="bullet">
<list-item>
<p>First, apply index-based permutation to the 3D cube using arrays <inline-formula>
<mml:math display="inline" id="im1">
<mml:mi>A</mml:mi>
</mml:math>
</inline-formula> and <inline-formula>
<mml:math display="inline" id="im2">
<mml:mi>B</mml:mi>
</mml:math>
</inline-formula> to reorder rows and columns, respectively. Specifically, perform row permutation as: <inline-formula>
<mml:math display="inline" id="im3">
<mml:mrow>
<mml:msub>
<mml:mstyle mathsize="normal">
<mml:mi>C</mml:mi>
</mml:mstyle>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mo>&#x2192;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mo>:</mml:mo>
<mml:mo>,</mml:mo>
<mml:mstyle mathsize="normal">
<mml:mi>A</mml:mi>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:mo>:</mml:mo>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula>
<mml:math display="inline" id="im4">
<mml:mrow>
<mml:msub>
<mml:mstyle mathsize="normal">
<mml:mi>C</mml:mi>
</mml:mstyle>
<mml:mi>G</mml:mi>
</mml:msub>
<mml:mo>&#x2192;</mml:mo>
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<list list-type="bullet">
<list-item>
<p>Next, complete the encryption by performing a bitwise XOR operation between the permuted 3D cube and the 3D key: <inline-formula>
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</p>
</list-item>
</list>
</sec>
<sec id="s3_4">
<label>3.4</label>
<title>Save the encrypted image and key</title>
<p>The encrypted 3D cube is converted back into a 2D color image by mapping the corresponding values to the RGB channels as shown in <xref ref-type="disp-formula" rid="eq20">Equations 20</xref>&#x2013;<xref ref-type="disp-formula" rid="eq22">22</xref>:</p>
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<label>(20)</label>
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<p>These channels are combined to generate the final encrypted image enc_img, which is saved to a file. The index arrays A, B, C, along with the key file key_bin, are stored for decryption.</p>
</sec>
<sec id="s3_5">
<label>3.5</label>
<title>Decryption process</title>
<p>To enable proper visualization and subsequent detection, the encrypted image must be decrypted. The decryption process involves four steps: load the key and encrypted image, reconstruct the 3D cube, decrypt each channel, and rebuild the 2D image.</p>
<sec id="s3_5_1">
<label>3.5.1</label>
<title>Load the key and encrypted image</title>
<p>The decryption process begins by loading the 3D key and index arrays (A, B, C) from the file system, along with the encrypted image file.</p>
</sec>
<sec id="s3_5_2">
<label>3.5.2</label>
<title>Reconstruct the 3D cube</title>
<p>Each color channel (R, G, B) of the image is reshaped into a 128 &#xd7; 128 &#xd7; 128 3D cube, where the first two dimensions represent pixel positions and the third dimension corresponds to the stacked image layers.</p>
</sec>
<sec id="s3_5_3">
<label>3.5.3</label>
<title>Decrypt each channel</title>
<p>For each color channel&#x2019;s 3D cube, two operations are applied:</p>
<list list-type="bullet">
<list-item>
<p>Bitwise XOR Restoration: The cube is first restored by applying a bitwise XOR operation with the original 3D key used during encryption.</p>
</list-item>
<list-item>
<p>Reverse Rotation: Then, reverse rotations are performed based on the index arrays A and B. Array A controls the reversal along rows, and B controls the columns. The rotation direction is opposite to the encryption process. The number of 90&#xb0; rotations is determined by the values in array C, applied in reverse order to maintain symmetry.</p>
</list-item>
</list>
</sec>
<sec id="s3_5_4">
<label>3.5.4</label>
<title>Rebuild the 2D image</title>
<p>The decrypted 3D cubes of the R, G, and B channels are converted back into 2D images. These channels are then merged to reconstruct the final color image, which is output as the decrypted result.</p>
</sec>
</sec>
</sec>
<sec id="s4">
<label>4</label>
<title>YOLOV11N-12D model</title>
<p>Deep learning-based object detection has demonstrated remarkable performance in disease recognition, with the YOLO series widely used for its speed and accuracy (<xref ref-type="bibr" rid="B7">Dai et&#xa0;al., 2022</xref>; <xref ref-type="bibr" rid="B47">Wang et&#xa0;al., 2024</xref>; <xref ref-type="bibr" rid="B50">Xie et&#xa0;al., 2024</xref>). However, traditional YOLO models struggle when dealing with small objects and class imbalance (<xref ref-type="bibr" rid="B37">Obu et&#xa0;al., 2023</xref>; <xref ref-type="bibr" rid="B1">A.N. and A.P., 2022</xref>). To address these issues, we propose an enhanced lightweight model, YOLOv11n-12D. As shown in <xref ref-type="fig" rid="f5">
<bold>Figure&#xa0;5</bold>
</xref>, the architecture consists of four components: Stem, Backbone, Neck, and Head. In our model, YOLOv11n serves as the student model, while YOLOv12s acts as the teacher. By leveraging knowledge distillation and detection loss, we enhance recall, reduce missed detections, and maintain efficiency, making it suitable for large-scale agricultural applications. The distillation process is detailed in <xref ref-type="fig" rid="f6">
<bold>Figure&#xa0;6</bold>
</xref>.</p>
<fig id="f5" position="float">
<label>Figure&#xa0;5</label>
<caption>
<p>Detection model architecture.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g005.tif">
<alt-text content-type="machine-generated">Flowchart diagram of a neural network architecture. It features three main sections: stem, backbone, and neck. Components include Conv3x3, SiLU, BN, Conv1x1 layers, MaxPool, up sampling, concat, and skip connections. Outputs are labeled head1, head2, and head3. A legend identifies layer types by color.</alt-text>
</graphic>
</fig>
<fig id="f6" position="float">
<label>Figure&#xa0;6</label>
<caption>
<p>Knowledge distillation process flow.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g006.tif">
<alt-text content-type="machine-generated">Flowchart illustrating the training process of a YOLO-based student model using a teacher model. It includes forward propagation, logit outputs, temperature scaling, softmax calculations, and loss calculations with early stopping and learning rate scheduling. Key elements like introduction of temperature parameter, focal and CIoU loss, and total loss calculation are depicted, leading to student model training with distillation loss.</alt-text>
</graphic>
</fig>
<p>The following are the steps of the distillation process:</p>
<list list-type="simple">
<list-item>
<p>a. The pre-trained YOLOv11n is used as the student model, and YOLOv12s as the teacher.  Z_<italic>t</italic> and <italic>Z</italic>_s are defined as shown in <xref ref-type="disp-formula" rid="eq23">Equations 23</xref>, <xref ref-type="disp-formula" rid="eq24">24</xref>: Augmented samples are input into both models to compute their respective logits:</p>
</list-item>
</list>
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<p>Since logits vary significantly in magnitude, a temperature parameter <italic>T = </italic>4.0 is applied to smooth them, stabilize gradients, and generate soft labels for distillation.</p>
<list list-type="simple">
<list-item>
<p>b. Temperature scaling (<italic>T = </italic>4.0) is applied to smooth the logits and obtain softened probability distributions for effective knowledge transfer as shown in <xref ref-type="disp-formula" rid="eq25">Equations 25</xref>, <xref ref-type="disp-formula" rid="eq26">26</xref>:</p>
</list-item>
</list>
<disp-formula id="eq25">
<label>(25)</label>
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<p>When <italic>T = </italic>1.0, the distribution reduces to standard softmax, limiting the ability to learn from the teacher. We adopt Kullback-Leibler divergence as the distillation loss: Distillation Loss = <italic>T</italic>
<sup>2</sup> &#xd7; KL(p <italic>
<sub>t</sub>
</italic> &#x2225; p<italic>
<sub>s</sub>
</italic>). Here, <italic>T</italic>
<sup>2</sup> offsets the gradient scaling effect caused by smoothing. A smaller KL value indicates better alignment between the student and teacher outputs, KL as defined in <xref ref-type="disp-formula" rid="eq27">Equation 27</xref>:</p>
<disp-formula id="eq27">
<label>(27)</label>
<mml:math display="block" id="M27">
<mml:mrow>
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</disp-formula>
<list list-type="simple">
<list-item>
<p>c. The student model is trained to balance knowledge from the teacher and performance on the original task. To achieve this, the detection loss and distillation loss are combined with a weighted sum: TotalLoss = (1 &#x2212; <italic>&#x3b1;</italic>) &#xd7; (Detection Loss) + <italic>&#x3b1;</italic> &#xd7; (Distillation Loss). Mixed precision training, learning rate scheduling, and early stopping strategies are employed to improve efficiency and convergence.</p>
</list-item>        <list-item>
<p>d. The detection loss consists of a weighted sum of classification loss (Focal Loss) and regression loss (CIoU Loss):Detection Loss=Focal Loss + CIoU Loss, A weight of <italic>&#x3b1; = </italic>0.7 is used to emphasize the distillation loss. The learning rate is adjusted using the OneCycleLR policy as shown in <xref ref-type="disp-formula" rid="eq28">Equation 28</xref>:</p>
</list-item>
</list>
<disp-formula id="eq28">
<label>(28)</label>
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<mml:mrow>
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</mml:mrow>
</mml:math>
</disp-formula>
<list list-type="simple">
<list-item>
<p>e. Training is terminated using an early stopping strategy when the condition specified in <xref ref-type="disp-formula" rid="eq29">Equation 29</xref> is met.</p>
</list-item>
</list>
<disp-formula id="eq29">
<label>(29)</label>
<mml:math display="block" id="M29">
<mml:mrow>
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</disp-formula>
<p>After each epoch,the validation set is evaluated using mAP, Precision, and Recall to monitor the effectiveness of the distillation strategy on student model performance.</p>
</sec>
<sec id="s5" sec-type="materials|methods">
<label>5</label>
<title>Materials preparation and optimization methods</title>
<p>To ensure high efficiency and accuracy in eggplant disease detection, a large-scale dataset was constructed, encompassing four categories: eggplant rot, fruit borer, healthy samples, and thrips. All images were annotated in YOLO format with standardized bounding boxes and precise class labels. The annotations were reviewed by agricultural experts to ensure high-quality and consistent labeling.The dataset was collected from multiple eggplant cultivation bases, encompassing various growth cycles, diverse lighting conditions, and all developmental stages of pest/disease infestation (from initial infection to characteristic symptom manifestation). Specifically, the test set comprises 745 representative images (containing 1516 annotated instances), while the remaining 7520 images were partitioned into training (5264 images) and validation (2256 images) sets at a 7:2:1 ratio. This scientifically designed partitioning scheme ensures both sufficient training data volume and reliable evaluation of model generalization capability. Sample differences between healthy and diseased eggplants are shown, highlighting the visual variability between categories (<xref ref-type="fig" rid="f7">
<bold>Figure&#xa0;7</bold>
</xref>). To address the limited quantity and variable quality of the collected raw images, we employed the Albumentations data augmentation library to enhance dataset diversity and improve model generalization and robustness (<xref ref-type="bibr" rid="B5">Buslaev et&#xa0;al., 2020</xref>; <xref ref-type="bibr" rid="B22">Korra et&#xa0;al., 2022</xref>). The overall workflow for material preparation and optimization is illustrated (<xref ref-type="fig" rid="f8">
<bold>Figure&#xa0;8</bold>
</xref>), and the specific data augmentation strategies are detailed as follows:</p>
<fig id="f7" position="float">
<label>Figure&#xa0;7</label>
<caption>
<p>
<bold>(A-D)</bold> are healthy eggplant images, <bold>(E-H)</bold> are diseased eggplant images.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g007.tif">
<alt-text content-type="machine-generated">A set of eight images labeled A to H depicting eggplants on plants. Images A to D show healthy eggplants marked with &#x201c;well-being&#x201d; scores ranging from 0.83 to 0.91. Images E to H display diseased eggplants with symptoms of pests and rot. E and G show eggplants affected by &#x201c;eggplant thrips.&#x201d; F highlights &#x201c;eggplant fruit borer&#x201d; damage. H shows multiple spots of &#x201c;eggplant rot&#x201d; with varying severity scores. Each image has colored boxes to identify and describe the condition of the eggplants.</alt-text>
</graphic>
</fig>
<fig id="f8" position="float">
<label>Figure&#xa0;8</label>
<caption>
<p>Diagram of materials analysis and optimization. <bold>(A)</bold> Focal loss. <bold>(B)</bold> CIoU loss. <bold>(C)</bold> Material preparation and optimization.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g008.tif">
<alt-text content-type="machine-generated">Flowchart illustrating three processes: A) Classification loss calculation using input prediction probability and ground truth label, involving cross-entropy loss, weight decay, and balancing factor. B) Regression loss calculation using predicted and ground truth boxes, involving Intersection over Union (IoU), center point distance penalty, and aspect ratio difference. C) Workflow from input data to model training, including dataset processing, annotation conversion to YOLO format, quality check, augmentation, and output results.</alt-text>
</graphic>
</fig>
<list list-type="bullet">
<list-item>
<p>Random Flip: Applies horizontal and vertical flips to simulate different viewing angles, enhancing feature recognition and robustness.</p>
</list-item>
<list-item>
<p>Color Jitter: Alters brightness, contrast, and saturation to mimic various lighting conditions.</p>
</list-item>
<list-item>
<p>Random Crop: Generates new samples by cropping image regions, helping reduce reliance on specific areas.</p>
</list-item>
<list-item>
<p>Random Rotation: Rotates images within &#xb1;30&#xb0; to improve viewpoint diversity and reduce angle bias.</p>
</list-item>
<list-item>
<p>Random Noise Addition: Introduces Gaussian or salt-and-pepper noise to improve performance under degraded image conditions.</p>
</list-item>
<list-item>
<p>Mosaic Augmentation: Merges multiple images to enrich contextual and visual diversity in complex scenes.</p>
</list-item>
<list-item>
<p>MixUp Augmentation: Blends two images and labels to promote smooth label transition and mitigate overfitting.</p>
</list-item>
</list>
<p>The applied augmentation techniques significantly enhance the model&#x2019;s robustness and generalization, allowing more reliable recognition of eggplant disease features under diverse conditions. Furthermore, the integration of Focal Loss and CloU Loss improves detection accuracy, achieving a final accuracy of 99.1% and effectively reducing the miss detection rate, thereby improving applicability in real-world agricultural scenarios.</p>
<sec id="s5_1">
<label>5.1</label>
<title>Focal loss</title>
<p>Focal Loss is designed to address class imbalance, especially in single-stage detectors like RetinaNet. Traditional cross-entropy is dominated by easy negatives, causing unstable training. Focal Loss introduces a modulation factor to focus learning on hard examples, improving detection of minority classes.The construction of this function is shown in part (a) of <xref ref-type="fig" rid="f8">
<bold>Figure&#xa0;8</bold>
</xref>. For a binary label: <italic>y</italic> &#x2208; {0,1}(0 for positive class and 1 for negative class), the predicted probability pt is defined as <xref ref-type="disp-formula" rid="eq30">Equation 30</xref>:</p>
<disp-formula id="eq30">
<label>(30)</label>
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<mml:mstyle mathsize="normal">
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<p>The standard cross-entropy loss is: CE = &#x2212;log(p<italic>
<sub>t</sub>
</italic>), Calculate the weight decay factor (1 &#x2212; p<italic>
<sub>t</sub>
</italic>)<italic>
<sup>&#x3b3;</sup>
</italic>, <italic>&#x3b3;</italic> is Hyperparameters, Typically takes values of 2 or 3, when <italic>p<sub>t</sub>
</italic> is small,the decay factor approaches 1 when p is small; otherwise, it approaches 0. A balancing factor is introduced to control the ratio of positive to negative samples. The final Focal Loss as presented in <xref ref-type="disp-formula" rid="eq31">Equation 31</xref>:</p>
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<label>(31)</label>
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<p>To compute the average Focal Loss for all samples in a batch, the classification loss is: <inline-formula>
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</p>
</sec>
<sec id="s5_2">
<label>5.2</label>
<title>CIoU loss</title>
<p>CIoU Loss is primarily used for object bounding box regression, especially in the context of rotated object detection. It optimizes the accuracy of the bounding box&#x2019;s location, size, and angle. CIoU Loss was introduced to improve the detection accuracy of rotated objects, as traditional bounding box regression methods typically only consider rectangular boxes, while CloU Loss also accounts for the angle of the rotated boxes. CloU Loss optimizes the bounding box regression of object detection models by considering the center point error, size error (width and height), and rotation angle error. The construction of this function is shown in part (b) of <xref ref-type="fig" rid="f8">
<bold>Figure&#xa0;8</bold>
</xref>. Calculate the Intersection over Union (IoU) between the predicted box b and the ground truth box <italic>b<sup>g</sup>
</italic>. As shown in <xref ref-type="disp-formula" rid="eq32">Equation 32</xref>:</p>
<disp-formula id="eq32">
<label>(32)</label>
<mml:math display="block" id="M32">
<mml:mrow>
<mml:mtext>IoU</mml:mtext>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mstyle mathsize="normal">
<mml:mi>I</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>t</mml:mi>
<mml:mi>e</mml:mi>
<mml:mi>r</mml:mi>
<mml:mi>s</mml:mi>
<mml:mi>e</mml:mi>
<mml:mi>c</mml:mi>
<mml:mi>t</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>n</mml:mi>
</mml:mstyle>
<mml:mtext>&#x2004;</mml:mtext>
<mml:mstyle mathsize="normal">
<mml:mi>A</mml:mi>
<mml:mi>r</mml:mi>
<mml:mi>e</mml:mi>
<mml:mi>a</mml:mi>
</mml:mstyle>
</mml:mrow>
<mml:mrow>
<mml:mstyle mathsize="normal">
<mml:mi>U</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>n</mml:mi>
</mml:mstyle>
<mml:mtext>&#x2004;</mml:mtext>
<mml:mstyle mathsize="normal">
<mml:mi>A</mml:mi>
<mml:mi>r</mml:mi>
<mml:mi>e</mml:mi>
<mml:mi>a</mml:mi>
</mml:mstyle>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>
<p>Calculate the Euclidean distance between the center of the predicted box and the center of the ground truth box, as defined in <xref ref-type="disp-formula" rid="eq33">Equation 33</xref>:</p>
<disp-formula id="eq33">
<label>(33)</label>
<mml:math display="block" id="M33">
<mml:mrow>
<mml:msup>
<mml:mi>&#x3c1;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>b</mml:mi>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>b</mml:mi>
<mml:mi>g</mml:mi>
</mml:msup>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mi>g</mml:mi>
</mml:msup>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>+</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>y</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>g</mml:mi>
</mml:msup>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</disp-formula>
<p>c can represent the diagonal distance of the smallest enclosing box that contains both the predicted box and the ground truth box as shown in <xref ref-type="disp-formula" rid="eq34">Equation 34</xref>: Distance penalty term:</p>
<disp-formula id="eq34">
<label>(34)</label>
<mml:math display="block" id="M34">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mi>&#x3c1;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo stretchy="false">(</mml:mo>
<mml:mstyle mathsize="normal">
<mml:mi>b</mml:mi>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mstyle mathsize="normal">
<mml:mi>b</mml:mi>
</mml:mstyle>
<mml:mi>g</mml:mi>
</mml:msup>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>
<p>Measure the difference in aspect ratio between the predicted box and the ground truth box, as defined in <xref ref-type="disp-formula" rid="eq35">Equation 35</xref>:</p>
<disp-formula id="eq35">
<label>(35)</label>
<mml:math display="block" id="M35">
<mml:mrow>
<mml:mi>V</mml:mi>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mn>4</mml:mn>
<mml:mrow>
<mml:msup>
<mml:mi>&#x3c0;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfrac>
<mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mtext>arctan&#xa0;</mml:mtext>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mi>w</mml:mi>
<mml:mi>g</mml:mi>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mi>h</mml:mi>
<mml:mi>g</mml:mi>
</mml:msup>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x2212;</mml:mo>
<mml:mtext>arctan&#xa0;</mml:mtext>
<mml:mfrac>
<mml:mi>w</mml:mi>
<mml:mi>h</mml:mi>
</mml:mfrac>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</disp-formula>
<p>Adaptive weight <italic>&#x3b1;</italic> is primarily used to balance the contribution of the aspect ratio loss term v in the overall CIoU loss. The result as shown in <xref ref-type="disp-formula" rid="eq36">Equation 36</xref>:</p>
<disp-formula id="eq36">
<label>(36)</label>
<mml:math display="block" id="M36">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mi>v</mml:mi>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mstyle mathsize="normal">
<mml:mi>I</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>U</mml:mi>
</mml:mstyle>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>
<p>By combining IoU, center distance, and aspect ratio, the final CIoU Loss is obtained as shown in <xref ref-type="disp-formula" rid="eq37">Equation 37</xref>:</p>
<disp-formula id="eq37">
<label>(37)</label>
<mml:math display="block" id="M37">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="script">L</mml:mi>
<mml:mrow>
<mml:mtext>CIoU</mml:mtext>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mtext>IoU</mml:mtext>
<mml:mo>+</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mi>&#x3c1;</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo stretchy="false">(</mml:mo>
<mml:mstyle mathsize="normal">
<mml:mi mathvariant="bold">b</mml:mi>
</mml:mstyle>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mstyle mathsize="normal">
<mml:mi mathvariant="bold">b</mml:mi>
</mml:mstyle>
<mml:mi>&#x261;</mml:mi>
</mml:msup>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mfrac>
<mml:mo>+</mml:mo>
<mml:mi>&#x3b1;</mml:mi>
<mml:mi>v</mml:mi>
</mml:mrow>
</mml:math>
</disp-formula>
</sec>
</sec>
<sec id="s6">
<label>6</label>
<title>Performance analysis</title>
<p>This section presents the evaluation metrics used to assess the performance of the two core components of the proposed system: the 3D chaotic cube-based encryption scheme for image security, and the YOLOv11n-12D-based detection model for eggplant disease diagnosis.</p>
<sec id="s6_1">
<label>6.1</label>
<title>Analysis of the proposed encryption scheme</title>
<p>To evaluate our encryption scheme, we developed a quantitative assessment system for image encryption security (<xref ref-type="table" rid="T1">
<bold>Table&#xa0;1</bold>
</xref>). It uses seven key indicators: contrast and mean square error (positively correlated) indicate pixel perturbation; information entropy shows randomness; while structural similarity (SSIM), energy value, homogeneity, and structural content (negatively correlated) assess structural damage, pattern concealment, and pixel disorder. This framework is based on research by <xref ref-type="bibr" rid="B12">Gupta and Chauhan (2021)</xref>; <xref ref-type="bibr" rid="B40">Rahman et&#xa0;al. (2025)</xref>, and <xref ref-type="bibr" rid="B20">Karmakar et&#xa0;al. (2021)</xref>.</p>
<table-wrap id="T1" position="float">
<label>Table&#xa0;1</label>
<caption>
<p>Evaluation parameters and their relation with image encryption security.</p>
</caption>
<table frame="hsides">
<thead>
<tr>
<th valign="middle" align="center">P.</th>
<th valign="middle" align="left">M.E.</th>
<th valign="middle" align="left">R.W.S.S.</th>
<th valign="middle" align="left">V.E.</th>
</tr>
</thead>
<tbody>
<tr>
<td valign="middle" align="center">Contrast</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im14">
<mml:mrow>
<mml:mtext>Contrast</mml:mtext>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mstyle displaystyle="true">
<mml:mo>&#x2211;</mml:mo>
</mml:mstyle>
<mml:mrow>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>|</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>b</mml:mi>
<mml:msup>
<mml:mo>|</mml:mo>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mtext>&#xa0;</mml:mtext>
<mml:mi>O</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">Contrast &#x221d; S.S</td>
<td valign="middle" align="left">Higher contrast reduces predictability and enhances security.</td>
</tr>
<tr>
<td valign="middle" align="center">SSIM</td>
<td valign="middle" align="center">
<inline-formula>
<mml:math display="inline" id="im15">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mi>S</mml:mi>
<mml:mi>I</mml:mi>
<mml:mi>M</mml:mi>
<mml:mo>=</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>2</mml:mn>
<mml:msub>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>x</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>y</mml:mi>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mi>C</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>2</mml:mn>
<mml:msub>
<mml:mi>&#x3c3;</mml:mi>
<mml:mrow>
<mml:mi>x</mml:mi>
<mml:mi>y</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mi>C</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">/</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:msubsup>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>x</mml:mi>
<mml:mn>2</mml:mn>
</mml:msubsup>
<mml:mo>+</mml:mo>
<mml:msubsup>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>y</mml:mi>
<mml:mn>2</mml:mn>
</mml:msubsup>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mi>C</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c3;</mml:mi>
<mml:mi>x</mml:mi>
<mml:mn>2</mml:mn>
</mml:msubsup>
<mml:mo>+</mml:mo>
<mml:msubsup>
<mml:mi>&#x3c3;</mml:mi>
<mml:mi>y</mml:mi>
<mml:mn>2</mml:mn>
</mml:msubsup>
<mml:mo>+</mml:mo>
<mml:msub>
<mml:mi>C</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im16">
<mml:mrow>
<mml:mtext>SSIM</mml:mtext>
<mml:mo>&#x221d;</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mtext>S</mml:mtext>
<mml:mo>.</mml:mo>
<mml:mtext>S</mml:mtext>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">Lower SSIM prevents structural leakage, improving security.</td>
</tr>
<tr>
<td valign="middle" align="center">MSE</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im17">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mi>S</mml:mi>
<mml:mi>E</mml:mi>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mi>N</mml:mi>
</mml:mfrac>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>N</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">MSE &#x221d; S.S</td>
<td valign="middle" align="left">Higher MSE increases difference, making decryption harder.</td>
</tr>
<tr>
<td valign="middle" align="center">Entropy</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im18">
<mml:mrow>
<mml:mtext>Entropy</mml:mtext>
<mml:mo>=</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mi>255</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>&#xd7;</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>log</mml:mi>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">Entropy &#x221d; S.S</td>
<td valign="middle" align="left">Higher entropy means more randomness, improving security.</td>
</tr>
<tr>
<td valign="middle" align="center">Energy</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im19">
<mml:mrow>
<mml:mtext>Energy</mml:mtext>
<mml:mo>=</mml:mo>
<mml:munder>
<mml:mstyle displaystyle="true">
<mml:mo>&#x2211;</mml:mo>
</mml:mstyle>
<mml:mrow>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:munder>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">[</mml:mo>
<mml:mi>O</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">]</mml:mo>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im20">
<mml:mrow>
<mml:mtext>Energy</mml:mtext>
<mml:mo>&#x221d;</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mtext>S</mml:mtext>
<mml:mo>.</mml:mo>
<mml:mtext>S</mml:mtext>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">Lower energy hides patterns, strengthening security.</td>
</tr>
<tr>
<td valign="middle" align="center">Homogeneity</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im21">
<mml:mrow>
<mml:mtext>Homogeneity</mml:mtext>
<mml:mo>=</mml:mo>
<mml:munder>
<mml:mstyle displaystyle="true">
<mml:mo>&#x2211;</mml:mo>
</mml:mstyle>
<mml:mrow>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:munder>
<mml:mfrac>
<mml:mrow>
<mml:mi>O</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>+</mml:mo>
<mml:mo>|</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>b</mml:mi>
<mml:mo>|</mml:mo>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im22">
<mml:mrow>
<mml:mtext>Homogeneity</mml:mtext>
<mml:mo>&#x221d;</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mtext>S</mml:mtext>
<mml:mo>.</mml:mo>
<mml:mtext>S</mml:mtext>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">Lower homogeneity increases pixel chaos, improving security.</td>
</tr>
<tr>
<td valign="middle" align="center">SC</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im23">
<mml:mrow>
<mml:mtext>SC</mml:mtext>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mtext>original&#x2004;</mml:mtext>
<mml:msup>
<mml:mrow>
<mml:mtext>image</mml:mtext>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mtext>original</mml:mtext>
<mml:mo>&#x2212;</mml:mo>
<mml:mtext>encrypted</mml:mtext>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">
<inline-formula>
<mml:math display="inline" id="im24">
<mml:mrow>
<mml:mtext>SC</mml:mtext>
<mml:mo>&#x221d;</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mtext>S</mml:mtext>
<mml:mo>.</mml:mo>
<mml:mtext>S</mml:mtext>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td valign="middle" align="left">Lower SC means less similarity to the original, enhancing security.</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn>
<p>P, Parameter; M.E., Mathematical Equation; R.W.S.S., Relationship with Strong Security (S.S); V.E., Variable Explanation.</p>
</fn>
</table-wrap-foot>
</table-wrap>
<p>
<xref ref-type="table" rid="T2">
<bold>Table&#xa0;2</bold>
</xref> shows that our method outperforms existing technologies (<xref ref-type="bibr" rid="B15">Huang et&#xa0;al., 2025</xref>; <xref ref-type="bibr" rid="B51">Xu et&#xa0;al., 2024</xref>; <xref ref-type="bibr" rid="B46">Ullah et&#xa0;al., 2025</xref>) in encryption quality, security, and efficiency. By integrating chaotic sequence generation, pixel permutation, and XOR encryption, our solution maintains consistent performance metrics for all test samples (both healthy and diseased eggplants). The entropy value is 7.6195 (close to the theoretical maximum of 8), and the pixel correlation coefficient is &#x2212;0.0084 (close to 0). Our method achieves high image fidelity (40.26 dB) and fast encryption speed (0.0127 seconds), which is 23 times faster than the fastest comparative method. It also preserves key features for disease identification, meeting smart agriculture&#x2019;s requirements for real-time performance, security, and feature preservation.</p>
<table-wrap id="T2" position="float">
<label>Table&#xa0;2</label>
<caption>
<p>Comparative performance analysis of eggplant image encryption methods.</p>
</caption>
<table frame="hsides">
<thead>
<tr>
<th valign="middle" colspan="9" align="center">Proposed work (encrypted images)</th>
</tr>
<tr>
<th valign="middle" align="left">Image type</th>
<th valign="middle" align="center">Homogeneity</th>
<th valign="middle" align="center">SC</th>
<th valign="middle" align="center">Entropy</th>
<th valign="middle" colspan="2" align="center">Correlation</th>
<th valign="middle" align="center">Energy</th>
<th valign="middle" align="center">Contrast</th>
<th valign="middle" align="center">Execution time (s)</th>
</tr>
</thead>
<tbody>
<tr>
<td valign="middle" align="left">Healthy 1</td>
<td valign="middle" align="center">0.0158</td>
<td valign="middle" align="center">0.6381</td>
<td valign="middle" align="center">7.6195</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0084</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">4905.8639</td>
<td valign="middle" align="center">0.0160</td>
</tr>
<tr>
<td valign="middle" align="left">Healthy 2</td>
<td valign="middle" align="center">0.0158</td>
<td valign="middle" align="center">0.8239</td>
<td valign="middle" align="center">7.6195</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0084</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">4905.8639</td>
<td valign="middle" align="center">0.0124</td>
</tr>
<tr>
<td valign="middle" align="left">Healthy 3</td>
<td valign="middle" align="center">0.0158</td>
<td valign="middle" align="center">0.4249</td>
<td valign="middle" align="center">7.6195</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0084</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">4905.8639</td>
<td valign="middle" align="center">0.0131</td>
</tr>
<tr>
<td valign="middle" align="left">Healthy 4</td>
<td valign="middle" align="center">0.0158</td>
<td valign="middle" align="center">0.3750</td>
<td valign="middle" align="center">7.6195</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0084</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">4905.8639</td>
<td valign="middle" align="center">0.0127</td>
</tr>
<tr>
<td valign="middle" align="left">Diseased 1</td>
<td valign="middle" align="center">0.0158</td>
<td valign="middle" align="center">
<bold>0.9212</bold>
</td>
<td valign="middle" align="center">7.6195</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0084</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">4905.8639</td>
<td valign="middle" align="center">
<bold>0.0111</bold>
</td>
</tr>
<tr>
<td valign="middle" align="left">Diseased 2</td>
<td valign="middle" align="center">0.0158</td>
<td valign="middle" align="center">0.5680</td>
<td valign="middle" align="center">7.6195</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0084</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">4905.8639</td>
<td valign="middle" align="center">0.0119</td>
</tr>
<tr>
<td valign="middle" align="left">Diseased 3</td>
<td valign="middle" align="center">0.0158</td>
<td valign="middle" align="center">0.6245</td>
<td valign="middle" align="center">7.6195</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0084</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">4905.8639</td>
<td valign="middle" align="center">0.0112</td>
</tr>
<tr>
<td valign="middle" align="left">Diseased 4</td>
<td valign="middle" align="center">0.0158</td>
<td valign="middle" align="center">0.6458</td>
<td valign="middle" align="center">7.6195</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0084</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">4905.8639</td>
<td valign="middle" align="center">0.0131</td>
</tr>
<tr>
<td valign="middle" align="left">Mean</td>
<td valign="middle" align="center">
<bold>0.0158</bold>
</td>
<td valign="middle" align="center">
<bold>0.6277</bold>
</td>
<td valign="middle" align="center">
<bold>7.6195</bold>
</td>
<td valign="middle" colspan="2" align="center">
<bold>-0.0084</bold>
</td>
<td valign="middle" align="center">
<bold>0.0001</bold>
</td>
<td valign="middle" align="center">
<bold>4905.8639</bold>
</td>
<td valign="middle" align="center">
<bold>0.0127</bold>
</td>
</tr>
</tbody>
</table>
<table frame="hsides">
<thead>
<tr>
<th valign="middle" colspan="9" align="center">Existing methods comparison</th>
</tr>
<tr>
<th valign="middle" align="left">Method</th>
<th valign="middle" align="center">MSE</th>
<th valign="middle" align="center">PSNR</th>
<th valign="middle" align="center">Entropy</th>
<th valign="middle" colspan="2" align="center">Correl.</th>
<th valign="middle" align="center">Energy</th>
<th valign="middle" align="center">Contrast</th>
<th valign="middle" align="center">Execution Time (s)</th>
</tr>
</thead>
<tbody>
<tr>
<td valign="middle" align="left">15</td>
<td valign="middle" align="center">5170.0723</td>
<td valign="middle" align="center">10.9958</td>
<td valign="middle" align="center">7.1346</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0707</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">2706.7328</td>
<td valign="middle" align="center">2.0395</td>
</tr>
<tr>
<td valign="middle" align="left">41</td>
<td valign="middle" align="center">8778.5593</td>
<td valign="middle" align="center">8.6966</td>
<td valign="middle" align="center">7.6232</td>
<td valign="middle" colspan="2" align="center">0.2083</td>
<td valign="middle" align="center">0.0001</td>
<td valign="middle" align="center">3970.8332</td>
<td valign="middle" align="center">0.2992</td>
</tr>
<tr>
<td valign="middle" align="left">45</td>
<td valign="middle" align="center">8256.4317</td>
<td valign="middle" align="center">8.9629</td>
<td valign="middle" align="center">
<bold>7.7475</bold>
</td>
<td valign="middle" colspan="2" align="center">&#x2212;0.0854</td>
<td valign="middle" align="center">0.0000</td>
<td valign="middle" align="center">
<bold>6158.4497</bold>
</td>
<td valign="middle" align="center">0.8220</td>
</tr>
<tr>
<td valign="middle" align="left">Ours</td>
<td valign="middle" align="center">
<bold>0.158</bold>
</td>
<td valign="middle" align="center">
<bold>40.2623</bold>
</td>
<td valign="middle" align="center">7.6195</td>
<td valign="middle" colspan="2" align="center">
<bold>-0.0084</bold>
</td>
<td valign="middle" align="center">
<bold>0.0001</bold>
</td>
<td valign="middle" align="center">4905.8639</td>
<td valign="middle" align="center">
<bold>0.0127</bold>
</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<p>Bold values are representative or key results.</p>
</table-wrap-foot>
</table-wrap>
</sec>
<sec id="s6_2">
<label>6.2</label>
<title>Key space analysis</title>
<p>The key space, representing the total number of possible keys, is a critical factor in resisting brute-force attacks. In this scheme, the user key is a 64-character hexadecimal string, corresponding to 256 bits. As each hex character encodes 4 bits, the key space size is: 2<sup>256</sup> &#x2248; 1.16 &#xd7; 10<sup>77</sup>. Such a vast key space is computationally infeasible to exhaust. Even at 10<sup>18</sup> keys per second, a brute-force search would take: <inline-formula>
<mml:math display="inline" id="im25">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mn>1.16</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mn>10</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>77</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mn>10</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>18</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>60</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>60</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>24</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>356</mml:mn>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#x2248;</mml:mo>
<mml:mn>3.67</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mn>10</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>50</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> year. These results confirm that the proposed key space is computationally infeasible to exhaust via brute-force attacks.</p>
</sec>
<sec id="s6_3">
<label>6.3</label>
<title>Attack resistance analysis</title>
<sec id="s6_3_1">
<label>6.3.1</label>
<title>Known-Plaintext attack</title>
<p>The proposed scheme uses a 3D Logistic Map, which exhibits strong sensitivity to initial conditions&#x2014;tiny variations lead to drastically different outputs. The chaotic system evolves as shown in <xref ref-type="disp-formula" rid="eq38">Equation 38</xref>:</p>
<disp-formula id="eq38">
<label>(38)</label>
<mml:math display="block" id="M38">
<mml:mrow>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>+</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mtext>r</mml:mtext>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>+</mml:mo>
<mml:mi>&#x3b2;</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>y</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>z</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</disp-formula>
<p>Chaotic behavior is verified by the Lyapunov exponent, as defined in <xref ref-type="disp-formula" rid="eq39">Equation 39</xref>:</p>
<disp-formula id="eq39">
<label>(39)</label>
<mml:math display="block" id="M39">
<mml:mrow>
<mml:mi>&#x3bb;</mml:mi>
<mml:mo>=</mml:mo>
<mml:munder>
<mml:mrow>
<mml:mi>lim</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>&#x2192;</mml:mo>
<mml:mi>&#x221e;</mml:mi>
</mml:mrow>
</mml:munder>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mi>n</mml:mi>
</mml:mfrac>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:mtext>ln&#xa0;</mml:mtext>
<mml:mo>|</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:mi>f</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>|</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<p>A positive exponent <italic>&#x3bb; &gt;</italic> 0, indicates exponential divergence. In our experiments, <italic>&#x3bb; = </italic>0.89 confirms high sensitivity, making it extremely difficult to reverse-engineer the key, even with known plaintext&#x2013;ciphertext pairs. This divergence is described as defined in <xref ref-type="disp-formula" rid="eq40">Equation 40</xref>:</p>
<disp-formula id="eq40">
<label>(40)</label>
<mml:math display="block" id="M40">
<mml:mrow>
<mml:mtext>&#x394;</mml:mtext>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mtext>&#x394;</mml:mtext>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#xb7;</mml:mo>
<mml:msup>
<mml:mtext>e</mml:mtext>
<mml:mrow>
<mml:mi>&#x3bb;</mml:mi>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</disp-formula>
<p>To initialize the chaotic system, we apply the SHA-256 hash function to the user key. With its strong collision resistance and irreversibility, the probability of a successful brute-force match is negligible, as defined in <xref ref-type="disp-formula" rid="eq41">Equation 41</xref>:</p>
<disp-formula id="eq41">
<label>(41)</label>
<mml:math display="block" id="M41">
<mml:mrow>
<mml:mi>P</mml:mi>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:msup>
<mml:mn>2</mml:mn>
<mml:mrow>
<mml:mn>256</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>
</sec>
<sec id="s6_3_2">
<label>6.3.2</label>
<title>Differential attack: NPCR</title>
<p>NPCR evaluates how a minor change in the input affects the encrypted output. It is defined as: <inline-formula>
<mml:math display="inline" id="im26">
<mml:mrow>
<mml:mtext>NPCR</mml:mtext>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mstyle displaystyle="true">
<mml:mo>&#x2211;</mml:mo>
</mml:mstyle>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mi>D</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>W</mml:mi>
<mml:mo>&#xd7;</mml:mo>
<mml:mi>H</mml:mi>
</mml:mrow>
</mml:mfrac>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>100</mml:mn>
<mml:mo>%</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>. <italic>D(i,j)</italic> as defined in <xref ref-type="disp-formula" rid="eq42">Equation 42</xref>:</p>
<disp-formula id="eq42">
<label>(42)</label>
<mml:math display="block" id="M42">
<mml:mrow>
<mml:mi>D</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mo>{</mml:mo>
<mml:mtable columnalign="left" equalrows="true" equalcolumns="true">
<mml:mtr columnalign="left">
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</disp-formula>
<p>Here, C<sub>1</sub>(<italic>i,j</italic>) and C<sub>2</sub>(<italic>i,j</italic>) denote the pixel values of two encrypted images with slight input differences. The ideal NPCR approaches 100%.</p>
</sec>
<sec id="s6_3_3">
<label>6.3.3</label>
<title>Differential attack: UACI</title>
<p>UACI quantifies the average intensity difference between two encrypted images, as shown in <xref ref-type="disp-formula" rid="eq43">Equation 43</xref>:</p>
<disp-formula id="eq43">
<label>(43)</label>
<mml:math display="block" id="M43">
<mml:mrow>
<mml:mtext>UACI</mml:mtext>
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<mml:mo>%</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<p>The ideal value should be close to 33%. Experimental results show: <italic>NPCR = </italic>99.63%<italic>, UACI = </italic>32.85%, These values confirm high resistance to differential attacks and strong sensitivity to input perturbations. <xref ref-type="fig" rid="f9">
<bold>Figure&#xa0;9</bold>
</xref> compares the pixel distribution: the original image (left) shows structured patterns, while the encrypted image (right) displays uniform randomness, demonstrating visual and statistical security.</p>
<fig id="f9" position="float">
<label>Figure&#xa0;9</label>
<caption>
<p>Pixel analysis diagram.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g009.tif">
<alt-text content-type="machine-generated">Left graph shows the pixel distribution of the original image with distinct peaks and valleys, indicating varied intensity levels. Right graph shows the pixel distribution of the encrypted image, revealing a uniform pattern with more evenly distributed pixel values. Both are presented in 3D plots with labeled axes.</alt-text>
</graphic>
</fig>
</sec>
</sec>
<sec id="s6_4">
<label>6.4</label>
<title>Statistical analysis</title>
<sec id="s6_4_1">
<label>6.4.1</label>
<title>Histogram analysis</title>
<p>
<xref ref-type="fig" rid="f10">
<bold>Figure&#xa0;10</bold>
</xref> illustrates the grayscale distributions of the original and encrypted images. The original image shows a clear peak in pixel intensity, while the encrypted image exhibits a nearly uniform distribution with no apparent structure. This indicates that the encryption process effectively randomizes the statistical properties of the original image, eliminating pixel concentration and preventing histogram-based attacks.</p>
<fig id="f10" position="float">
<label>Figure&#xa0;10</label>
<caption>
<p>Image grayscale distribution.</p>
</caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpls-16-1634408-g010.tif">
<alt-text content-type="machine-generated">Two histograms are shown side by side. The left graph is titled &#x201c;Histogram of the original image&#x201d; with a prominent peak at the start and a flat distribution after. The right graph, labeled &#x201c;Encrypted image Histogram,&#x201d; displays a more uniform distribution with a bell-shaped curve, peaking around the center.</alt-text>
</graphic>
</fig>
</sec>
<sec id="s6_4_2">
<label>6.4.2</label>
<title>Pixel autocorrelation analysis</title>
<p>The Pearson correlation coefficient measures the linear relationship between adjacent pixel values and defined as in <xref ref-type="disp-formula" rid="eq44">Equation 44</xref>:</p>
<disp-formula id="eq44">
<label>(44)</label>
<mml:math display="block" id="M44">
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mo>=</mml:mo>
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</mml:mrow>
</mml:msqrt>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>
<p>Ideally, <italic>r</italic> &#x2248; 0 indicates no correlation. In our experiment, <italic>r&#xa0;=</italic>0.005, confirming that the encrypted image lacks linear pixel dependencies, which enhances its resistance to statistical attacks.</p>
</sec>
</sec>
<sec id="s6_5">
<label>6.5</label>
<title>Detection performance evaluation</title>
<p>This study evaluated the YOLOv11n-12D model using <xref ref-type="table" rid="T3">
<bold>Tables&#xa0;3</bold>
</xref>, <xref ref-type="table" rid="T4">
<bold>4</bold>
</xref>, confirming its innovative breakthroughs. <xref ref-type="table" rid="T3">
<bold>Table&#xa0;3</bold>
</xref> highlights the model&#x2019;s superior performance in detecting four eggplant diseases: rot (mAP@0.5=0.861), fruit borer (0.872), healthy plants (0.911), and notably thrips (0.753), a 6.5% improvement over the baseline. It achieves accuracy comparable to YOLOv12s (gap &lt;2%) via knowledge distillation while remaining lightweight. <xref ref-type="table" rid="T4">
<bold>Table&#xa0;4</bold>
</xref> provides a comprehensive performance comparison. The model maintains near-teacher accuracy (1.2% mAP@0.5 difference) and achieves a 2.7ms inference speed&#x2014;3.6&#xd7; faster than YOLOv12s (9.6ms) and 3.2&#xd7; faster than YOLOv8n (8.7ms). Its F1-Score (0.804) outperforms YOLOv10n (0.764) and YOLOv8n (0.785), with a 4.5% improvement in the stricter mAP@0.5:0.95 metric, demonstrating stability in multi-scale detection.</p>
<table-wrap id="T3" position="float">
<label>Table&#xa0;3</label>
<caption>
<p>Performance comparison of YOLO models for eggplant disease detection.</p>
</caption>
<table frame="hsides">
<thead>
<tr>
<th valign="middle" rowspan="2" align="left">Model</th>
<th valign="middle" rowspan="2" align="center">Category</th>
<th valign="middle" colspan="2" align="center">Dataset</th>
<th valign="middle" colspan="4" align="center">Performance metrics</th>
</tr>
<tr>
<th valign="middle" align="center">Images</th>
<th valign="middle" align="center">Instances</th>
<th valign="middle" align="center">Precision</th>
<th valign="middle" align="center">Recall</th>
<th valign="middle" align="center">mAP@0.5</th>
<th valign="middle" align="center">mAP@0.5:0.95</th>
</tr>
</thead>
<tbody>
<tr>
<td valign="middle" rowspan="4" align="left">YOLOv11n-12D</td>
<td valign="middle" align="left">Eggplant Rot</td>
<td valign="middle" align="center">117</td>
<td valign="middle" align="center">544</td>
<td valign="middle" align="center">0.838</td>
<td valign="middle" align="center">0.814</td>
<td valign="middle" align="center">
<bold>0.861</bold>
</td>
<td valign="middle" align="center">0.565</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Fruit</td>
<td valign="middle" align="center">237</td>
<td valign="middle" align="center">323</td>
<td valign="middle" align="center">
<bold>0.861</bold>
</td>
<td valign="middle" align="center">0.819</td>
<td valign="middle" align="center">
<bold>0.872</bold>
</td>
<td valign="middle" align="center">0.407</td>
</tr>
<tr>
<td valign="middle" align="left">Healthy</td>
<td valign="middle" align="center">216</td>
<td valign="middle" align="center">331</td>
<td valign="middle" align="center">0.871</td>
<td valign="middle" align="center">0.886</td>
<td valign="middle" align="center">
<bold>0.911</bold>
</td>
<td valign="middle" align="center">0.725</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Thrips</td>
<td valign="middle" align="center">175</td>
<td valign="middle" align="center">318</td>
<td valign="middle" align="center">0.752</td>
<td valign="middle" align="center">0.643</td>
<td valign="middle" align="center">
<bold>0.753</bold>
</td>
<td valign="middle" align="center">0.442</td>
</tr>
<tr>
<td valign="middle" rowspan="4" align="left">YOLOv11n-12D<break/>Unencrypted<break/>(Baseline)</td>
<td valign="middle" align="left">Eggplant Rot</td>
<td valign="middle" align="center">117</td>
<td valign="middle" align="center">544</td>
<td valign="middle" align="center">0.838</td>
<td valign="middle" align="center">0.814</td>
<td valign="middle" align="center">
<bold>0.861</bold>
</td>
<td valign="middle" align="center">0.565</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Fruit</td>
<td valign="middle" align="center">237</td>
<td valign="middle" align="center">323</td>
<td valign="middle" align="center">0.861</td>
<td valign="middle" align="center">0.819</td>
<td valign="middle" align="center">
<bold>0.872</bold>
</td>
<td valign="middle" align="center">0.407</td>
</tr>
<tr>
<td valign="middle" align="left">Healthy</td>
<td valign="middle" align="center">216</td>
<td valign="middle" align="center">331</td>
<td valign="middle" align="center">0.871</td>
<td valign="middle" align="center">0.886</td>
<td valign="middle" align="center">
<bold>0.911</bold>
</td>
<td valign="middle" align="center">0.725</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Thrips</td>
<td valign="middle" align="center">175</td>
<td valign="middle" align="center">318</td>
<td valign="middle" align="center">0.752</td>
<td valign="middle" align="center">0.643</td>
<td valign="middle" align="center">
<bold>0.753</bold>
</td>
<td valign="middle" align="center">0.442</td>
</tr>
<tr>
<td valign="middle" rowspan="4" align="left">yolov11n</td>
<td valign="middle" align="left">Eggplant Rot</td>
<td valign="middle" align="center">117</td>
<td valign="middle" align="center">544</td>
<td valign="middle" align="center">0.838</td>
<td valign="middle" align="center">0.724</td>
<td valign="middle" align="center">0.822</td>
<td valign="middle" align="center">0.552</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Fruit</td>
<td valign="middle" align="center">237</td>
<td valign="middle" align="center">323</td>
<td valign="middle" align="center">0.826</td>
<td valign="middle" align="center">0.807</td>
<td valign="middle" align="center">0.850</td>
<td valign="middle" align="center">0.395</td>
</tr>
<tr>
<td valign="middle" align="left">Healthy</td>
<td valign="middle" align="center">216</td>
<td valign="middle" align="center">331</td>
<td valign="middle" align="center">0.861</td>
<td valign="middle" align="center">0.891</td>
<td valign="middle" align="center">
<bold>0.924</bold>
</td>
<td valign="middle" align="center">0.728</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Thrips</td>
<td valign="middle" align="center">175</td>
<td valign="middle" align="center">318</td>
<td valign="middle" align="center">0.810</td>
<td valign="middle" align="center">0.575</td>
<td valign="middle" align="center">0.707</td>
<td valign="middle" align="center">0.411</td>
</tr>
<tr>
<td valign="middle" rowspan="4" align="left">yolov12s</td>
<td valign="middle" align="left">Eggplant Rot</td>
<td valign="middle" align="center">117</td>
<td valign="middle" align="center">544</td>
<td valign="middle" align="center">0.852</td>
<td valign="middle" align="center">0.827</td>
<td valign="middle" align="center">0.873</td>
<td valign="middle" align="center">0.582</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Fruit</td>
<td valign="middle" align="center">237</td>
<td valign="middle" align="center">323</td>
<td valign="middle" align="center">0.879</td>
<td valign="middle" align="center">0.831</td>
<td valign="middle" align="center">0.884</td>
<td valign="middle" align="center">0.418</td>
</tr>
<tr>
<td valign="middle" align="left">Healthy</td>
<td valign="middle" align="center">216</td>
<td valign="middle" align="center">331</td>
<td valign="middle" align="center">
<bold>0.883</bold>
</td>
<td valign="middle" align="center">0.901</td>
<td valign="middle" align="center">0.922</td>
<td valign="middle" align="center">
<bold>0.741</bold>
</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Thrips</td>
<td valign="middle" align="center">175</td>
<td valign="middle" align="center">318</td>
<td valign="middle" align="center">0.765</td>
<td valign="middle" align="center">0.659</td>
<td valign="middle" align="center">0.767</td>
<td valign="middle" align="center">0.456</td>
</tr>
<tr>
<td valign="middle" rowspan="4" align="left">yolov10n</td>
<td valign="middle" align="left">Eggplant Rot</td>
<td valign="middle" align="center">117</td>
<td valign="middle" align="center">544</td>
<td valign="middle" align="center">0.805</td>
<td valign="middle" align="center">0.706</td>
<td valign="middle" align="center">0.781</td>
<td valign="middle" align="center">0.502</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Fruit</td>
<td valign="middle" align="center">237</td>
<td valign="middle" align="center">323</td>
<td valign="middle" align="center">0.821</td>
<td valign="middle" align="center">0.768</td>
<td valign="middle" align="center">0.826</td>
<td valign="middle" align="center">0.384</td>
</tr>
<tr>
<td valign="middle" align="left">Healthy</td>
<td valign="middle" align="center">216</td>
<td valign="middle" align="center">331</td>
<td valign="middle" align="center">0.852</td>
<td valign="middle" align="center">
<bold>0.904</bold>
</td>
<td valign="middle" align="center">0.910</td>
<td valign="middle" align="center">0.719</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Thrips</td>
<td valign="middle" align="center">175</td>
<td valign="middle" align="center">318</td>
<td valign="middle" align="center">0.770</td>
<td valign="middle" align="center">0.505</td>
<td valign="middle" align="center">0.672</td>
<td valign="middle" align="center">0.397</td>
</tr>
<tr>
<td valign="middle" rowspan="4" align="left">yolov8n</td>
<td valign="middle" align="left">Eggplant Rot</td>
<td valign="middle" align="center">117</td>
<td valign="middle" align="center">544</td>
<td valign="middle" align="center">0.820</td>
<td valign="middle" align="center">0.750</td>
<td valign="middle" align="center">0.817</td>
<td valign="middle" align="center">0.524</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Fruit</td>
<td valign="middle" align="center">237</td>
<td valign="middle" align="center">323</td>
<td valign="middle" align="center">0.859</td>
<td valign="middle" align="center">0.794</td>
<td valign="middle" align="center">0.857</td>
<td valign="middle" align="center">0.408</td>
</tr>
<tr>
<td valign="middle" align="left">Healthy</td>
<td valign="middle" align="center">216</td>
<td valign="middle" align="center">331</td>
<td valign="middle" align="center">0.815</td>
<td valign="middle" align="center">
<bold>0.907</bold>
</td>
<td valign="middle" align="center">0.917</td>
<td valign="middle" align="center">0.717</td>
</tr>
<tr>
<td valign="middle" align="left">Eggplant Thrips</td>
<td valign="middle" align="center">175</td>
<td valign="middle" align="center">318</td>
<td valign="middle" align="center">0.781</td>
<td valign="middle" align="center">0.566</td>
<td valign="middle" align="center">0.697</td>
<td valign="middle" align="center">0.399</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<p>Bold values are representative or key results.</p>
</table-wrap-foot>
</table-wrap>
<table-wrap id="T4" position="float">
<label>Table&#xa0;4</label>
<caption>
<p>Comprehensive performance comparison.</p>
</caption>
<table frame="hsides">
<thead>
<tr>
<th valign="middle" align="left">Model version</th>
<th valign="middle" align="center">Precision</th>
<th valign="middle" align="center">Recall</th>
<th valign="middle" align="center">mAP@0.5</th>
<th valign="middle" align="center">mAP@0.5:0.95</th>
<th valign="middle" align="center">F1-score</th>
<th valign="middle" align="center">Inference speed (ms)</th>
</tr>
</thead>
<tbody>
<tr>
<td valign="middle" align="left">YOLOv11n-12D</td>
<td valign="middle" align="center">0.831</td>
<td valign="middle" align="center">0.791</td>
<td valign="middle" align="center">0.849</td>
<td valign="middle" align="center">0.535</td>
<td valign="middle" align="center">0.804</td>
<td valign="middle" align="center">
<bold>2</bold>.<bold>7</bold>
</td>
</tr>
<tr>
<td valign="middle" align="left">YOLOv11n</td>
<td valign="middle" align="center">0.834</td>
<td valign="middle" align="center">0.749</td>
<td valign="middle" align="center">0.826</td>
<td valign="middle" align="center">0.522</td>
<td valign="middle" align="center">0.789</td>
<td valign="middle" align="center">3.3</td>
</tr>
<tr>
<td valign="middle" align="left">YOLOv12s</td>
<td valign="middle" align="center">
<bold>0.845</bold>
</td>
<td valign="middle" align="center">
<bold>0.804</bold>
</td>
<td valign="middle" align="center">
<bold>0.861</bold>
</td>
<td valign="middle" align="center">
<bold>0.549</bold>
</td>
<td valign="middle" align="center">
<bold>0.812</bold>
</td>
<td valign="middle" align="center">9.6</td>
</tr>
<tr>
<td valign="middle" align="left">YOLOv10n</td>
<td valign="middle" align="center">0.812</td>
<td valign="middle" align="center">0.721</td>
<td valign="middle" align="center">0.797</td>
<td valign="middle" align="center">0.501</td>
<td valign="middle" align="center">0.764</td>
<td valign="middle" align="center">3.1</td>
</tr>
<tr>
<td valign="middle" align="left">YOLOv8n</td>
<td valign="middle" align="center">0.819</td>
<td valign="middle" align="center">0.754</td>
<td valign="middle" align="center">0.822</td>
<td valign="middle" align="center">0.512</td>
<td valign="middle" align="center">0.785</td>
<td valign="middle" align="center">8.7</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<p>Bold values are representative or key results.</p>
</table-wrap-foot>
</table-wrap>
<p>Comprehensive analysis of data from both tables demonstrates that YOLOv11n-12D, through the synergistic optimization of knowledge distillation and Focal Loss, successfully overcomes the traditional trade-off between accuracy and efficiency, achieving the innovative breakthrough of &#x201c;teacher-level accuracy with edge-level efficiency&#x201d; and providing reliable technical support for real-time disease detection in smart agriculture.</p>
</sec>
<sec id="s6_6">
<label>6.6</label>
<title>Information entropy analysis</title>
<p>The formula for information entropy as defined in <xref ref-type="disp-formula" rid="eq45">Equation 45</xref>:</p>
<disp-formula id="eq45">
<label>(45)</label>
<mml:math display="block" id="M45">
<mml:mrow>
<mml:mtext>H</mml:mtext>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>X</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:munderover>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mi>255</mml:mi>
</mml:munderover>
</mml:mstyle>
<mml:mstyle mathsize="normal">
<mml:mi mathvariant="bold">p</mml:mi>
</mml:mstyle>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>log</mml:mi>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mstyle mathsize="normal">
<mml:mi>p</mml:mi>
</mml:mstyle>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>x</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>&#xa0;</mml:mo>
<mml:mo>&#xa0;</mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mstyle mathsize="normal">
<mml:mi>I</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>f</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>r</mml:mi>
<mml:mi>m</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>t</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>n</mml:mi>
</mml:mstyle>
<mml:mtext>&#x2004;</mml:mtext>
<mml:mstyle mathsize="normal">
<mml:mi>e</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>t</mml:mi>
<mml:mi>r</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>y</mml:mi>
</mml:mstyle>
<mml:mtext>&#x2004;</mml:mtext>
<mml:mo stretchy="false">(</mml:mo>
<mml:mstyle mathsize="normal">
<mml:mi>u</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>t</mml:mi>
</mml:mstyle>
<mml:mo>:</mml:mo>
<mml:mtext>&#x2004;</mml:mtext>
<mml:mstyle mathsize="normal">
<mml:mi>b</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>t</mml:mi>
</mml:mstyle>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
<p>The entropy of the original image ranges from 5 to 7, while the encrypted image&#x2019;s entropy is close to 8 (theoretical maximum), showing a more uniform and random pixel distribution. Our calculated entropy value is 7.6195, indicating high information entropy, which helps prevent information leakage and statistical analysis attacks.</p>
</sec>
</sec>
<sec id="s7" sec-type="conclusions">
<label>7</label>
<title>Conclusion</title>
<p>This paper proposes an integrated system for eggplant disease detection that combines image encryption and deep learning-based recognition. The system employs a lightweight encryption scheme based on 3D chaotic mapping and pixel permutation to secure image transmission with low computational overhead. It then utilizes an optimized YOLOv11n-12D model to process the decrypted images, achieving high detection accuracy and real-time performance. A teacher&#x2013;student knowledge distillation strategy is incorporated to further enhance model robustness. Experimental results demonstrate that the system not only safeguards data privacy but also outperforms existing methods in accuracy, speed, and stability, offering a reliable solution for smart agriculture. At the same time,our future research will focus on enabling disease detection directly on encrypted images to eliminate the risk of data leakage during decryption. This will involve exploring privacy-preserving techniques like homomorphic encryption and designing lightweight models that can operate effectively in the encrypted domain.</p>
</sec>
<sec id="s8" sec-type="discussion">
<label>8</label>
<title>Discussion</title>
<p>Our study proposes a framework that integrates image encryption with deep learning based object detection for real time, privacy-preserving crop disease monitoring. The designed 3D chaotic cube encryption scheme demonstrates strong security performance, achieving high entropy (7.6195), low pixel correlation (0.0084), and strong resistance to statistical and differential attacks (NPCR = 99.63%, UACI = 32.85%). Meanwhile, the YOLOv11n-12D model retains the detection performance of the teacher model while achieving fast inference speed (2.7 ms), with a notable mAP improvement of +6.5% in small-object detection such as eggplant thrips. This solution offers a promising approach for advancing smart agriculture in rural or resource limited areas. By encrypting images before transmission and decrypting them only during model inference, the framework strikes a practical balance between data security and operational efficiency. Its compatibility with edge devices further supports deployment in real world scenarios, where data privacy, bandwidth limitations, and low computing resources are common challenges. Despite the promising results, the current framework still requires decryption before detection, which introduces a temporary risk of data exposure. Future work will focus on privacy preserving deep learning techniques that support inference directly in the encrypted domain, such as homomorphic encryption or secure multi party computation. Further validation on larger and more diverse crop datasets is also needed to assess generalization. Enhancing the interpretability of both the detection model and the encryption process will help improve transparency and user trust in practical applications.</p>
</sec>
</body>
<back>
<sec id="s9" sec-type="data-availability">
<title>Data availability statement</title>
<p>The original contributions presented in the study are included in the article/supplementary material. Further inquiries can be directed to the corresponding author.</p>
</sec>
<sec id="s10" sec-type="author-contributions">
<title>Author contributions</title>
<p>JH: Writing &#x2013; original draft, Funding acquisition, Methodology. ZW: Writing &#x2013; original draft. YD: Writing &#x2013; review &amp; editing, Conceptualization. YG: Data curation, Writing &#x2013; original draft. RF: Investigation, Writing &#x2013; review &amp; editing, Supervision.</p>
</sec>
<sec id="s11" sec-type="funding-information">
<title>Funding</title>
<p>The author(s) declare financial support was received for the research and/or publication of this article. This research was supported by the Weifang University of Science and Technology A-Class Doctoral Research Fund (Grant No. KJRC2024014), Shan dong Provincial Natural Science Foundation (Grant No. ZR2025QC649), and the Weifang University of Science and Technology A-Class Doctoral Research Fund (Grant No. KJRC2024006).</p>
</sec>
<sec id="s12" sec-type="COI-statement">
<title>Conflict of interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec id="s13" sec-type="ai-statement">
<title>Generative AI statement</title>
<p>The author(s) declare that no Generative AI was used in the creation of this manuscript.</p>
<p>Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.</p>
</sec>
<sec id="s14" sec-type="disclaimer">
<title>Publisher&#x2019;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
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