Modified Chaotic Map and DNA Operators Strengthen Digital Image Encryption Scheme

Computer scientists have designed a digital image encryption scheme that combines a modified one-dimensional chaotic map with multiple DNA computing operators to protect visual data during storage and electronic transmission. Developed by Muhammad Akraam, Bushra Zafar, Muhammad Kashif Iqbal, and Tabasam Rashid, the cryptographic framework addresses critical security vulnerabilities that arise when classical text-based algorithms such as the Advanced Encryption Standard, International Data Encryption Algorithm, or linear feedback shift registers are applied directly to graphical files. Digital images possess high data redundancy and exceptionally strong correlation among adjacent pixels, characteristics that render traditional block ciphers computationally inefficient and structurally fragile against statistical cryptanalysis. To overcome these constraints, the researchers engineered an integrated encryption pipeline that subjects image data to chaotic pixel scrambling, dynamic nucleotide substitution, and binary-level diffusion, creating an unreadable ciphertext that thoroughly conceals underlying visual structures.

The foundation of the proposed cryptographic system is a newly constructed modified logistic map, designed to overcome the classical trade-offs inherent in chaotic systems. While multi-dimensional and hyperchaotic systems offer broad parameter spaces and intricate nonlinear attractors, their mathematical complexity demands heavy computational processing and presents operational challenges on resource-constrained hardware. Conversely, conventional one-dimensional chaotic maps, such as the standard logistic map, offer lightweight execution and structural simplicity but suffer from narrow chaotic parameter windows, periodic interruptions, and predictable statistical distributions that leave them vulnerable to differential attacks. The research team formulated an algebraic modification expressed as X(t+1) equals the quantity u times X(t) times 1 minus X(t), minus the quantity 1 minus X(t), all modulo 1, where u represents the control parameter and X(1) serves as the initial state within the unit interval. This formulation successfully preserves the low computational overhead of one-dimensional systems while extending the functional parameter interval across an exceptionally broad range.

Dynamical system evaluations confirmed that the modified logistic map exhibits superior chaotic properties compared to traditional models. Bifurcation analysis demonstrated that while the conventional logistic map enters chaotic motion only when its parameter exceeds approximately 3.57 and collapses entirely when the parameter surpasses 4.0, the modified map achieves chaotic behavior at a much lower threshold of u greater than 1.2 and sustains continuous chaotic dynamics even at extreme values up to 5 times 10 to the twelfth power. Complementary evaluations using Lyapunov exponents revealed sustained positive values over this expanded parameter space, verifying that initial trajectory paths diverge exponentially over time and provide heightened sensitivity to initial conditions. Further mathematical validation using the 0–1 test for chaos yielded asymptotic growth rate values approaching 1, while sample entropy assessments conducted across diverse parameter configurations consistently exceeded 1.5 and 2, confirming that the generated time series possess high complexity, negligible pattern repetition, and robust pseudo-random qualities required for key stream generation.

The procedural workflow of the cryptosystem begins by converting an input grayscale plaintext image into a one-dimensional array before deploying chaotic sequences to govern subsequent transformations. Three distinct parameter pairs iterate the modified map to yield floating-point chaotic trajectories, which are scaled, amplified by a factor of 10 to the tenth power, and mapped modulo 256 to generate three discrete positive integer key streams. These key streams dynamically direct the translation of standard 8-bit image pixels into four-character DNA nucleotide sequences adhering strictly to Watson-Crick biological pairing rules. Among the 24 possible permutations for assigning two-bit binary pairs to adenine, cytosine, guanine, and thymine, only eight configurations preserve complementary Watson-Crick base pairing. The encryption framework incorporates all eight valid biological coding tables, using the first chaotic key stream to dictate dynamically which specific encoding rule converts each individual byte into its corresponding nucleotide string.

Following nucleotide transformation, the system executes an advanced DNA-level diffusion stage that simultaneously harnesses four distinct biological arithmetic operations. Rather than relying on a static algebraic mechanism, the encryption pipeline draws on a second chaotic key sequence to choose dynamically among nucleotide addition, subtraction, bitwise exclusive-OR, and nucleotide complementation for every consecutive position. These operations are evaluated through formal modular mapping functions that mirror biological interactions across the four nucleic bases. By continuously shifting between addition, subtraction, complementary inversion, and exclusive-OR, the algorithm conceals which specific mathematical operator was applied to alter any given pixel group, preventing unauthorized observers from tracing algebraic relations through the cryptographic layer. The diffused nucleotides are subsequently decoded back into decimal integers using rules selected by the third chaotic keystream, ensuring multi-tiered algorithmic unpredictability.

The cryptographic pipeline then advances into binary-level diffusion and image permutation stages designed to eliminate spatial dependencies. The intermediate decimal sequence is combined with large secret positive integer constants reaching up to 10 to the fifteenth power, converted into 8-bit binary representations, and subjected to bitwise exclusive-OR operations against the primary chaotic keystream. To disrupt spatial correlations, a fourth modified logistic map sequence is generated using secret control parameters that are dynamically perturbed by the arithmetic sum of the diffused image pixels scaled by 10 to the negative fifth power. Sorting this chaotic sequence creates an index permutation vector that rearranges the physical pixel coordinates without changing their collective sum, allowing legitimate recipients to recover the exact update parameters from the ciphertext without transmitting per-round keys. Finally, the scrambled one-dimensional array is reconstructed into the original two-dimensional image dimensions, completing the forward encryption cycle.

Experimental validation of the cryptosystem was executed using MATLAB 2018b on standard computing hardware equipped with an Intel Core i5 processor and 8 gigabytes of memory, analyzing standard benchmark images including Lena, Baboon, House, Pepper, and Cameraman obtained from the USC-SIPI Image Database. Key space analysis revealed that the collective parameter configurations—encompassing four floating-point control parameters, four initial conditions, and two large auxiliary integers—yield an immense key space of 10 to the 132nd power. This capacity substantially surpasses the standard cryptographic threshold of 2 to the 100th power, establishing absolute resilience against brute-force attacks. Histogram analysis of the resulting ciphertexts displayed flat, uniformly distributed pixel frequency profiles across all 256 intensity values, with Chi-square test scores consistently falling below the critical threshold of 293.2478 at a 0.05 significance level, proving that the encrypted outputs retain no discernible statistical clues regarding the original images.

Extensive information theory and correlation assessments further demonstrated the system’s ability to disguise visual information as random noise. Global information entropy values across encrypted test images approached the theoretical optimum of 8 for an 8-bit grayscale space, while Local Shannon Entropy scores fell squarely within the optimal confidence interval of 7.901901305 to 7.903037329. Analysis of adjacent pixel correlation coefficients across horizontal, vertical, and diagonal directions revealed that strong plaintext correlations approaching 1 dropped to near-zero values following encryption, effectively severing statistical dependencies between contiguous pixels. Furthermore, image distortion metrics showed high Mean Squared Error and Mean Absolute Deviation values alongside low Peak Signal-to-Noise Ratio figures, while Structural Similarity Index Measure scores approached zero, demonstrating total visual disruption and structural concealment between plaintext and ciphertext images.

Sensitivity analyses confirmed that the algorithm provides powerful resistance against differential cryptanalysis and chosen-plaintext attacks. Evaluating the Number of Changing Pixel Rate and the Unified Average Changing Intensity following single-pixel modifications yielded values near the theoretical ideals of 99.6 percent and 33.5 percent, respectively, proving that a minute change in the input cascades unpredictably across the entire encrypted output. Similar testing under slight variations of the secret key parameters generated completely scrambled images during decryption, verifying extreme key sensitivity. Encrypting entirely black and entirely white synthetic images likewise yielded noise-like ciphertexts with ideal entropy and sensitivity scores, demonstrating that adversaries cannot exploit known or chosen inputs to extract operational key material or reconstruct image contents.

Statistical evaluation using the National Institute of Standards and Technology test suite confirmed the randomness of binary sequences generated by both the standalone modified chaotic map and the final encrypted images, with resulting p-values successfully meeting standard significance benchmarks. In addressing practical implementation, the authors noted that the architecture is designed intentionally as a lossless encryption scheme prioritizing exact decryption recovery rather than resilience to transmission channel noise. Because the pervasive diffusion mechanisms cause single-bit ciphertext errors to disperse globally into severe decryption distortions, the authors emphasized that channel-induced noise and transmission losses must be resolved using separate transport-layer protocols, such as forward error correction or retransmission strategies, allowing the encryption framework to dedicate its full computational strength to cryptographic confidentiality.

Subject of Research: Digital image encryption using a modified one-dimensional chaotic map and multi-operator DNA diffusion

Article Title: DNA-Based image encryption scheme employing multiple DNA operators simultaneously and a modified chaotic Map

Article References: Akraam, M., Zafar, B., Iqbal, M. K., & Rashid, T. (2026). DNA-Based image encryption scheme employing multiple DNA operators simultaneously and a modified chaotic Map. Discover Computing, 29(1), Article 865. https://doi.org/10.1007/s10791-026-10808-4

Image Credits: AI Generated

DOI: 10.1007/s10791-026-10808-4

Keywords: image encryption, DNA cryptography, chaotic systems, modified logistic map, cybersecurity, pixel diffusion, cryptanalysis, information entropy, DNA-Based, image, encryption, scheme