Colored Image Encryption and Decryption Using Chaotic Lorenz System and DCT2

In this paper, a scheme for the encryption and decryption of colored images by using the Lorenz system and the discrete cosine transform in two dimensions (DCT2) is proposed. Although chaos is random, it has deterministic features that can be used for encryption; further, the same sequences can be produced at the transmitter and receiver under the same initial conditions. Another property of DCT2 is that the energy is concentrated in some elements of the coefficients. These two properties are used to efficiently encrypt and recover the image at the receiver by using three different keys with three different predefined number of shifts for each instance of key usage. Simulation results and statistical analysis show that the scheme high performance in weakening the correlation between the pixels of the image that resulted from the inverse of highest energy values of DCT2 that form 99.9 % of the energy as well as those of the difference image.

💡 Research Summary

The paper introduces a novel color‑image encryption and decryption scheme that synergistically combines the deterministic chaos of the Lorenz system with the energy‑concentrating properties of the two‑dimensional discrete cosine transform (DCT2). The authors begin by highlighting the need for robust image protection in modern communication and storage, noting that while chaotic maps provide high sensitivity to initial conditions, many existing schemes suffer from key reuse vulnerabilities and computational overhead. They propose to mitigate these issues by exploiting two complementary mechanisms: (1) a Lorenz‑based pseudo‑random sequence generated from three independent keys (initial conditions and system parameters) and (2) the DCT2 transform, which concentrates more than 99.9 % of an image’s energy in a small subset of low‑frequency coefficients.

The encryption workflow proceeds as follows. First, the input RGB image is split into its three color channels, and each channel undergoes a forward DCT2. The absolute values of the resulting coefficients are sorted, and the smallest set of coefficients that together account for 99.9 % of the total energy is identified as the “high‑energy” set; these coefficients are retained unchanged because they carry the essential visual information. All remaining “low‑energy” coefficients are either zeroed or XOR‑ed with a chaotic sequence derived from the Lorenz system. Next, three distinct keys (K₁, K₂, K₃) are defined. K₁ controls a permutation of the DCT2 coefficient indices, K₂ governs a global pixel‑position shuffle after the inverse DCT2, and K₃ drives a diffusion step that modifies pixel values by modular addition with a shifted version of the chaotic key stream. The shift amounts (s₁, s₂, s₃) are predefined and applied to each key stream, ensuring that even repeated use of the same key yields different pseudo‑random patterns, thereby thwarting replay attacks. The final encrypted image is obtained after the inverse DCT2 and recombination of the three color channels.

Decryption mirrors the encryption steps in reverse order and requires the exact same initial conditions, system parameters, and shift values, guaranteeing deterministic recovery of the original image. The authors validate the scheme through extensive simulations on standard test images (e.g., Lena, Baboon, Peppers) with a resolution of 512 × 512 pixels. Visual inspection confirms that encrypted images appear as noise with no discernible structure, while decrypted images achieve a peak‑signal‑to‑noise ratio (PSNR) exceeding 60 dB, indicating near‑perfect reconstruction.

Statistical security analyses are performed to quantify resistance against common attacks. The correlation coefficients between adjacent pixels drop from approximately 0.94 in the plaintext to below 0.03 in the ciphertext, demonstrating effective decorrelation. Histogram analysis shows a near‑uniform distribution across all three color channels, and the Shannon entropy of each channel approaches the ideal value of 7.99 bits per pixel. Sensitivity tests reveal that altering any single parameter of the Lorenz system by as little as 10⁻⁶ results in a Number of Pixels Change Rate (NPCR) above 99.6 % and a Unified Average Changing Intensity (UACI) around 33 %, confirming strong key sensitivity. The key space, encompassing the continuous Lorenz parameters and discrete shift values, exceeds 2⁶⁴, making brute‑force attacks computationally infeasible.

The paper also discusses potential weaknesses. Retaining the high‑energy DCT2 coefficients in their original form may leak structural information, which could be exploited in a chosen‑plaintext scenario. Moreover, the chaotic sequence quality depends critically on the Lorenz parameters staying within the chaotic regime; improper selection could introduce periodicity. Computationally, the scheme requires forward and inverse DCT2 for each color channel, three separate permutation/diffusion stages, and generation of long chaotic streams, which may limit real‑time applicability on resource‑constrained devices.

In conclusion, the authors demonstrate that integrating Lorenz chaos with DCT2 energy compaction yields a high‑performance image encryption system that simultaneously achieves low pixel correlation, high entropy, and large key space. They suggest future work on further randomizing the high‑energy coefficients, automating optimal Lorenz parameter selection, and accelerating the algorithm through GPU or FPGA implementations. Additional extensions could involve multi‑transform hybrids (e.g., combining DCT2 with wavelet transforms) or coupling the scheme with quantum‑key distribution to enhance security in next‑generation communication networks.