💻 computer science

A Grayscale image encryption using cubic fractional transformation and logistic map

This paper proposes a grayscale image encryption scheme that combines a cubic fractional transformation-generated S-box with a logistic map to perform substitution, diffusion, and XOR-based scrambling, demonstrating robust security through comprehensive statistical and key sensitivity analyses.

Original authors: Abdul Qadeer, Rashid Ali, Tahir Sajjad Ali

Published 2026-07-22

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CC BY 4.0

Original authors: Abdul Qadeer, Rashid Ali, Tahir Sajjad Ali

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the digital world as a bustling city where data is the currency, and images are the most valuable jewels being traded. In this high-speed exchange, keeping those jewels safe from thieves is a constant battle. This is the world of cryptography, the science of secret writing. At its heart lies a simple but powerful idea: take something recognizable, like a photo of your face, and scramble it so thoroughly that it looks like static noise to anyone without the right key. To do this, cryptographers use two main tricks. First, they use substitution, which is like swapping every letter in a word for a completely different symbol according to a secret codebook. Second, they use scrambling, which is like taking a deck of cards, shuffling them until the order is chaotic, and then dealing them out. The goal is to make the final result look so random that no pattern can be found, ensuring that even if a hacker steals the message, they can't make sense of it without the specific instructions to unscramble it.

In the paper titled "A Grayscale image encryption using cubic fractional transformation and logistic map," researchers Abdul Qadeer, Rashid Ali, and Tahir Sajjad Ali propose a new, three-step recipe for scrambling grayscale images (black-and-white photos) to make them incredibly hard to crack. They don't just use one trick; they combine a custom-made codebook, a mathematical shuffling machine, and a chaotic generator to create a fortress around the image.

First, the team builds a custom S-box (Substitution-box), which acts as their secret codebook. Instead of using a standard or random codebook, they construct this one using a mathematical formula called a Cubic Fractional Transformation (CFT). Think of this as baking a unique, one-of-a-kind cipher wheel where every number from 0 to 255 (representing the shades of gray in a photo) is swapped for a new, unpredictable number. The authors suggest that this specific mathematical recipe creates a codebook that is highly non-linear and resistant to common attacks, with an average non-linearity of 106.75.

Once the pixel values are swapped using this custom codebook, the image is still in the same shape, just with different colors. The next step is permutation, or shuffling. The researchers use the same CFT math to generate two random sequences. These sequences act like a set of instructions to perform circular shifts—imagine sliding the rows and columns of the image in a circle, like a sliding puzzle, but in a way that is determined by the secret keys. This step moves the pixels around so that a pixel that was originally next to its neighbor is now far away, breaking up any local patterns.

Finally, the team applies a scrambling process using a Logistic map, a famous mathematical function known for producing chaotic, unpredictable sequences. They use this to generate a stream of random numbers. Then, they perform a Boolean XOR operation (a specific type of binary math) between the shuffled image and this random stream. This is the final lock: it mixes the image data with the chaos, ensuring that even a tiny change in the original image or the key results in a completely different final image.

The researchers tested their method on standard test images like "Lena," "Baboon," and "Clock." Their simulations show that the resulting encrypted images look like pure static noise. When they measured the entropy (a measure of randomness), the encrypted images scored very close to the ideal maximum of 8 (specifically around 7.997), suggesting the data is highly random. They also checked the key sensitivity, finding that changing the secret key by a tiny amount (like adding 0.000000000000005) made it impossible to recover the original image, proving the system is extremely sensitive to the correct key. Furthermore, the NPCR (Number of Pixel Change Rate) and UACI (Unified Average Changing Intensity) values were high (around 99.6% and 33.4% respectively), indicating that changing just one pixel in the original photo causes a massive, unpredictable change in the encrypted version.

The paper suggests that this combination of a CFT-generated S-box, CFT-based shuffling, and Logistic map scrambling creates a robust encryption scheme. The authors conclude that their method resists common cryptographic attacks, such as trying to guess the key or analyzing the statistical patterns of the image. However, these findings are based on simulations and mathematical analysis of the algorithm's performance on test images, not on real-world deployment against active hackers. The work proposes a secure method for protecting digital images, offering a new way to lock down visual data in an era where privacy is increasingly precious.

Technical Summary: Grayscale Image Encryption Using Cubic Fractional Transformation and Logistic Map

Problem Statement
In the modern technological era, the rapid evolution of information technology necessitates robust methods for protecting data, particularly image-based information. While various encryption techniques exist, including symmetric and asymmetric key systems, there is a continuous need for efficient algorithms that ensure secure transmission by converting recognizable images into unrecognizable forms. Existing methods often rely on chaotic systems or algebraic structures, but the authors identify a need for schemes that effectively combine substitution, permutation, and diffusion properties to resist common cryptographic attacks such as differential, linear, and statistical analysis.

Methodology
The paper proposes a novel grayscale image encryption scheme that integrates a Substitution-box (S-box) generated via Cubic Fractional Transformation (CFT) with a Logistic map. The encryption process for a $256 \times 256$ grayscale image follows a three-stage workflow:

S-Box Generation and Substitution:
- An S-box is constructed using the Cubic Fractional Transformation (CFT). The transformation is defined as $F(x) = \frac{1}{ax^3 + b} \mod (2n+1)$ , with specific parameters $a=95$ and $b=15$ .
- To handle invalid outputs (specifically when the denominator is zero or results fall outside the valid range), the authors apply a modified CFT that assigns specific values (0 and 106) to problematic inputs ( $x=176$ and $x=184$ ).
- This generated S-box is used to substitute the pixel values of the original image, creating a substituted image matrix $S'$ .
Permutation (Diffusion):
- Two random sequences are generated using the CFT with different parameter sets ( $k_1$ and $k_2$ ) to drive the permutation process.
- The algorithm applies a column-wise circular shift to the substituted image $S'$ using the first sequence, followed by a row-wise circular shift using the second sequence. This results in a pre-encrypted image $P$ .
Scrambling (Confusion):
- A Logistic map, defined by $t_{n+1} = \gamma \times t_n (1 - t_n)$ , is utilized to generate a chaotic sequence.
- The sequence is processed to eliminate transient effects and converted into 8-bit integer values.
- A bitwise XOR operation is performed between the pre-encrypted image pixels and the chaotic sequence. The process employs a feedback mechanism where the current encrypted pixel depends on the previous encrypted pixel, the current pre-encrypted pixel, and the chaotic sequence value.

Decryption
The decryption algorithm reverses the process:

The XOR operation is reversed using the same chaotic sequence generated by the Logistic map.
The row-wise and column-wise circular shifts are reversed using the CFT-generated sequences.
The inverse S-box (derived from the original CFT parameters) is applied to restore the original pixel values.

Key Contributions

Novel S-Box Construction: The paper claims this is the first method to utilize Cubic Fractional Transformation specifically for constructing an S-box for image encryption, extending previous work on Linear Fractional Transformations.
Hybrid Approach: The scheme effectively combines the confusion properties of an algebraically constructed S-box with the diffusion capabilities of CFT-based permutations and the chaotic randomness of the Logistic map.
Comprehensive Security Analysis: The authors provide a rigorous evaluation of the proposed scheme against standard cryptographic metrics.

Results and Performance
The algorithm was tested on standard grayscale images (Clock, Lena, Baboon, Pepper, Female) using MATLAB R2017a. The results indicate:

Key Space: The scheme utilizes four secret keys ( $k_0, k_1, k_2, k_3$ ), resulting in a key space of $2^{147}$ , which exceeds the minimum requirement to resist brute-force attacks.
Key Sensitivity: The system is highly sensitive to key changes; a minute alteration (e.g., $0.000000000000005$) in the Logistic map parameter $\gamma$ results in a completely different decrypted image, preventing unauthorized access.
Differential Analysis: The proposed method achieves high NPCR (Number of Pixel Change Rate) and UACI (Unified Average Changing Intensity) values. For example, the Lena image yielded an NPCR of 99.5910% and UACI of 33.3255%, comparable to or exceeding existing schemes, indicating strong resistance to known and chosen plaintext attacks.
Correlation Coefficients: The encryption significantly reduces the correlation between adjacent pixels. While plain images show high correlation (e.g., 0.9451 for Lena horizontal), the cipher images show values close to zero (e.g., -0.0049), effectively breaking statistical relationships.
Entropy: The information entropy of the encrypted images approaches the ideal value of 8 (e.g., 7.9973 for Lena), indicating high randomness and unpredictability.
Histogram Analysis: The histograms of the encrypted images exhibit a uniform distribution, showing no statistical hints of the original image structure.

Significance and Claims
The authors claim that the proposed algorithm is a secure and effective method for grayscale image encryption. They assert that the specific S-box constructed via CFT possesses strong cryptographic properties, including high non-linearity (average 106.75), low linear probability (0.1484), and low differential probability (0.0391), making it resistant to linear and differential cryptanalysis. The study concludes that the combination of CFT-based substitution/permutation and Logistic map-based scrambling creates a robust system capable of resisting common cryptographic attacks, suitable for encrypting images of various types and sizes.

Technical Summary: Grayscale Image Encryption Using Cubic Fractional Transformation and Logistic Map

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