Adaptive Fuzzy Logic and Hybrid Chaotic Maps Secure Digital Image Encryption

A content-dependent image encryption framework developed by researchers Biswarup Yogi, Raj Majumdar, Soham Modak, Aritra Biswas, Umashankar Rawat, and Satyabrata Roy combines Mamdani fuzzy inference with multi-stage chaotic permutation, partial Advanced Encryption Standard substitution, and lightweight diffusion to protect digital imagery across networked environments. Described in Discover Internet of Things, the architecture addresses spatial redundancy and strong adjacent pixel correlations inherent in uncompressed digital pictures, which frequently render conventional text-oriented ciphers computationally burdensome or structurally vulnerable. Rather than relying on rigid cryptographic parameters or unvarying diffusion schedules, the deterministic framework extracts image-specific statistical properties to regulate chaotic initialization, substitution probability, and diffusion strength dynamically. Extensive experimental evaluations across seven standard SIPI benchmark images demonstrated that the scheme achieved Number of Pixel Change Rate values spanning 99.67 percent to 99.89 percent, Unified Average Changing Intensity values from 33.11 percent to 33.66 percent, and information entropy scores between 7.984 and 7.995 bits. Furthermore, execution on standard personal computing platforms required average encryption and decryption durations of 0.84 seconds and 0.79 seconds, respectively.

The engineering motivation behind the framework arises from the rapid expansion of digital imaging across cloud storage repositories, medical diagnostic platforms, surveillance monitoring grids, and Internet of Things environments where image security remains critical. Standard cryptographic schemes such as DES and AES were primarily developed for one-dimensional textual streams, and their direct application to multidimensional image matrices introduces unnecessary computational overhead while failing to exploit specific spatial characteristics. Previous chaos-based permutation and diffusion systems attempted to overcome these constraints, yet many architectures incorporated fixed control parameters, standalone chaotic systems, or exhaustive substitution layers that restricted dynamic responsiveness and increased computational strain on resource-constrained hardware. A single chaotic mapping can exhibit inadequate pseudo-randomness, whereas complete non-linear substitution over entire pixel matrices significantly escalates latency on embedded hardware. To overcome these limitations, the research team developed an entropy-driven control mechanism connecting the inherent statistical characteristics of an input plaintext image directly to downstream cryptographic transformations, achieving an operational trade-off among confusion complexity, spatial diffusion capability, arithmetic efficiency, and exact deterministic reversibility.

The operational workflow processes an input colour image by downscaling the spatial grid to reduce computational complexity while retaining essential structural features. Feature extraction modules subsequently calculate three primary statistical indicators across the downscaled matrix: Shannon information entropy, arithmetic mean intensity, and standard deviation contrast. The calculated entropy quantifies the initial intensity distribution, the mean measures average luminance, and the contrast evaluates intensity spread across spatial coordinates. These quantitative values serve as crisp inputs to a specialized Mamdani fuzzy inference system comprising three input variables and three output variables. Within the fuzzy inference engine, input variables are partitioned into Low, Medium, and High linguistic categories using triangular membership functions for intermediate regions and trapezoidal membership functions across boundary domains to ensure continuous transitions. The evaluated statistical features are mapped through a curated linguistic rule base that formalizes non-linear relationships between visual statistics and cryptographic requirements, employing minimum logic for conjunctions and maximum logic for disjunctions across the active rule set.

Defuzzification within the Mamdani system occurs through centroid integration, which independently resolves the aggregated fuzzy output sets into crisp numerical values representing chaotic seed scaling, substitution probability, and diffusion intensity. The derived chaotic seed scaling factor is combined with private scalar information to compute a key-dependent chaotic seed, establishing a mathematical bridge between plaintext statistics and subsequent pseudorandom sequence generation. This content-dependent seed initializes a conditional multi-level chaotic permutation stage applied independently across the red, green, and blue colour channels. The spatial permutation combines a KAA chaotic map with a hybrid logistic–sine chaotic system. The KAA mapping generates a chaotic sequence that is converted via sorting indices into a permutation vector, which rearranges spatial coordinates whenever the operational encryption configuration activates the KAA routine. By conditionally applying the KAA stage, the architecture provides flexible structural scrambling without compromising the underlying mathematical reversibility essential during cipher decryption.

Following conditional KAA scrambling, pixel matrices undergo a secondary hierarchical shuffling stage governed by a coupled hybrid logistic–sine map. This hybrid mapping integrates the non-linear dynamic behavior of sinusoidal functions and logistic formulations to yield unpredictable chaotic trajectories within bounded phase spaces. Generating permutation indices through ascending-order coordinate sorting, the hybrid sequence thoroughly redistributes pixel positions across the channel grids, dismantling coherent spatial structures and substantially mitigating vulnerability to visual or statistical pattern reconstruction. Once chaotic permutation completes, the rearranged channel data enters a deterministic selective non-linear substitution phase utilizing the standard AES substitution box. Unlike conventional cryptographic algorithms that execute uniform S-box transformations across all coordinate values, the proposed framework employs a key-dependent deterministic mask generated by a pseudorandom number generator evaluated against the fuzzy substitution probability. Pixels whose generated mask value falls below the calculated threshold undergo non-linear S-box substitution, whereas remaining pixels bypass the operation, providing robust confusion without incurring exhaustive computational costs.

The final forward encryption stage applies an adaptive lightweight diffusion operation governed by the fuzzy diffusion strength parameter to establish global dependency among neighboring pixels. Operating in a raster-scan sequence across each colour channel, a key-dependent keystream is generated by compounding chaotic states, the fuzzy diffusion parameter, and secret key elements through iterative bitwise exclusive-OR operations. Each pixel is XORed with the corresponding keystream element and the previously processed ciphertext pixel, ensuring that localized perturbations propagate continuously across the entire coordinate array. The individual diffused colour channels are subsequently concatenated into a unified cipher matrix. The decryption architecture precisely mirrors these transformations in strict reverse order to guarantee lossless plaintext recovery. Ciphertext arrays are split into respective colour channels, and inverse XOR diffusion recursive routines reverse the spatial propagation. Deterministic mask regeneration subsequently directs inverse AES S-box substitution, followed by inverse hybrid logistic–sine and inverse KAA coordinate permutations, successfully restoring original plaintext dimensions and pixel intensities without arithmetic distortion.

Quantitative validation of the proposed architecture was conducted through extensive experiments using seven standard SIPI benchmark test images, evaluating Mean Squared Error, Peak Signal-to-Noise Ratio, differential sensitivity metrics, and spatial randomness. Across thirty independent execution cycles, encrypted images yielded elevated Mean Squared Error values reaching 8662.93 and low Peak Signal-to-Noise Ratio values around 7.98 decibels, demonstrating substantial visual degradation and perceptual dissimilarity between plaintext and ciphertext representations. Differential cryptanalysis resistance was confirmed by Number of Pixel Change Rate values approaching 99.89 percent and Unified Average Changing Intensity values near 33.66 percent. These differential metrics indicate that altering a single bit in the original plaintext image produces widespread variations across nearly all ciphertext pixel values, confirming strong avalanche propagation. Global information entropy evaluations yielded values approaching 7.995 bits out of an ideal 8-bit ceiling, while non-overlapping local block entropy analysis consistently confirmed values near 8 bits, demonstrating uniform local randomness and negligible spatial leakage.

Statistical evaluation further incorporated histogram analysis, spatial correlation assessments, mutual information calculations, and NIST SP 800-22 randomness verification to confirm resilience against cryptanalytic attacks. Plaintext images such as Baboon and Airplane displayed highly uneven, modality-dense histogram profiles, whereas the corresponding encrypted outputs exhibited flat, uniform distributions confirmed by low chi-square test statistics. Spatial correlation assessments measuring dependencies between adjacent pixel pairs in horizontal, vertical, and diagonal orientations revealed that strong plaintext correlations approaching unity were reduced to near-zero values within approximately plus or minus 0.01 in the cipher images. Mutual information metrics between plaintext and ciphertext remained below 0.014 bits, verifying negligible statistical dependency. Keystreams subjected to the rigorous NIST SP 800-22 statistical test suite satisfied the required acceptance criteria with pass rates ranging from 98.5 percent to 99.6 percent, verifying that generated binary streams are indistinguishable from true random sequences.

Security assessments also examined key space magnitude, key sensitivity, and non-linear dynamic behavior within the constituent chaotic generators. In accordance with Kerckhoffs principle, auxiliary metadata encompassing downscaled spatial dimensions and fuzzy control parameters are assumed visible to an adversary without compromising underlying confidentiality, as plaintext reconstruction strictly mandates the secret cryptographic keys. The effective secret key space combines a 256-bit elliptic-curve cryptography scalar with channel-dependent chaotic initialization states providing an additional uncertainty of 2^{100} to 2^{104} states, establishing a total key space ranging from 2^{356} to 2^{360}, which comfortably exceeds the threshold necessary to defeat brute-force exhaustion attacks. Furthermore, dynamic evaluations through Lyapunov exponent calculations yielded positive values exceeding 0.82 for both the KAA and hybrid logistic–sine maps, while bifurcation and phase-space trajectories demonstrated extended chaotic intervals free from periodic degradation, confirming high sensitivity to initial values and key configurations.

The authors contextualized their findings against existing chaotic schemes and outlined clear operational boundaries regarding deployment platforms and imaging domains. While comparative literature evaluations demonstrated that the proposed scheme attained superior NPCR values and competitive entropy scores, the authors emphasized that published comparisons rely on differing dataset parameters and implementation environments rather than synchronized head-to-head benchmarking. Furthermore, the integration of Mamdani fuzzy inference with multi-stage chaotic operations introduces computational overhead relative to minimal static ciphers. Although successfully executed on personal computing hardware and ARM-based IoT systems, current evaluations remain software-based rather than deployed within dedicated hardware ASICs or large-scale edge networks. Additionally, because experimental validation relied on standard benchmark images rather than domain-specific medical imaging repositories, claims regarding clinical applicability represent potential suitability rather than direct empirical validation, motivating planned future research focused on FPGA acceleration, hyperspectral data, and medical image modalities.

Subject of Research: Deterministic adaptive image encryption using fuzzy logic and chaotic systems

Article Title: Deterministic Mamdani fuzzy-controlled KAA and hybrid logistic–sine chaotic image encryption with AES S-box substitution

Article References: Yogi, B., Majumdar, R., Modak, S., Biswas, A., Rawat, U., & Roy, S. (2026). Deterministic Mamdani fuzzy-controlled KAA and hybrid logistic–sine chaotic image encryption with AES S-box substitution. Discover Internet of Things, 6(1), Article 186. https://doi.org/10.1007/s43926-026-00500-w

Image Credits: AI Generated

DOI: 10.1007/s43926-026-00500-w

Keywords: image encryption, chaotic cryptography, fuzzy inference, Mamdani model, AES S-box, information security, Internet of Things, differential cryptanalysis, Deterministic, Mamdani, fuzzy-controlled, hybrid