Knight's Tour Research Articles

Data scrambler knight tour algorithm

Nowadays, data scrambling remains a vital technique to protect sensitive information by shuffling it in a way that makes it difficult to decipher or reverse-engineer while still maintaining its usability for legitimate purposes. As manipulating the usability of the scrambled data remains a challenge on the background of risking losing data and getting them re-identified by attackers, scrambling and descrambling should be accomplished faster by not increasing data loss and re-identification risks. A scrambling algorithm must have a linear time complexity, still shuffling the data to minimize the risks further. A promising approach is based on the knight open tour problem, whose solutions appear like a random series of knight positions. Hence, a knight open tour algorithm is formalized, by which the knight seems to move chaotically across the chessboard. The formalization is presented as an indented pseudocode to implement it efficiently, whichever programming language is used. The output is a square matrix representing the knight open tour. Based on the knight tour matrix, data scrambler and descrambler algorithms are presented in the same manner. The algorithms have a linear time complexity. The knight-tour scrambling has a sufficiently low guess probability if an appropriate depth of scrambling is used, where the data is re-scrambled repetitively. The scrambling depth is determined by repetitive application of the chessboard matrix, whose size usually increases as the scrambling is deepened. Compared to the pseudorandom shuffling of the data along with storing the shuffled indices, the knight-tour descrambling key is stored and sent far simpler yet ensures proper data security.

Novel grayscale image encryption based on 4D fractional-order hyperchaotic system, 2D Henon map and knight tour algorithm

With the increasing frequency of data exchange, the security of transmitted information, especially images, has become paramount. This paper proposes a novel algorithm for encrypting grayscale images of any dimension by using a proposed fractional-order (FO) 4D hyperchaotic system, 2D Henon chaotic map permutation, and the knight tour algorithm. Initially, chaotic sequences are generated by utilizing the proposed FO 4D hyperchaotic system, which are later employed to rearrange and shuffle the entire image pixels to bolster the efficacy of image encryption. To introduce an additional layer of diffusion, 2D Henon chaotic map permutation is used. Furthermore, the knight tour algorithm is applied by starting from a chosen point and executing specified rounds on the scrambled image to increase the encryption’s robustness. The resultant image encryption algorithm undergoes thorough testing and evaluation. It exhibits high sensitivity to the encryption key and boasts a larger key space, rendering it more resistant to brute-force attacks. The proposed algorithm demonstrates an approximate correlation of 0 between adjacent pixels. Further, encryption of a grayscale image of size 256 × 256 takes approximately 0.4 seconds, rendering it more suitable for cryptographic purposes.

PVD with knight tour algorithm

PVD with knight tour algorithm

An algorithm based on 6D fractional order hyperchaotic system and knight tour algorithm to encrypt image

Abstract The security guarantee of data transmission is becoming more crucial as the frequency of information interchange rises. Ensuring the security of images is essential since they serve as a vital transmission medium. This research suggests an image encryption method that combines the knight tour algorithm with a 6D fractional order hyperchaotic system. First, chaotic sequences are produced using a fractional order hyperchaotic system, which is then utilized to index order and jumble the entire image. To retrieve the image after the second scrambling, choose the knight tour beginning point and run ten rounds of knight tour algorithms on the scrambled image. Thirdly, to maximize the efficiency of picture encryption, employ diffusion methods. The outcomes of the imaging experiment were lastly tested and assessed. The security of the image can be successfully guaranteed by a high-dimensional fractional order hyperchaotic system. This is because its high dimensionality gives it a larger key space than the low dimensional system. This is why it can resist attacks more effectively. After a series of evaluation experiments, it is obvious that this encryption scheme has good encryption performance.

Proving the existence of Euclidean knight’s tours on n × n × ⋯ × n chessboards for n < 4

The Knight's Tour problem consists of finding a Hamiltonian path for the knight on a given set of points so that the knight can visit exactly once every vertex of the mentioned set. In the present, we provide a 5-dimensional alternative to the well-known statement that it is not ever possible for a knight to visit once every vertex of $C(3,k):=\{\underbrace{\{0,1,2\} \times \{0,1,2\}\times \cdots \times \{0,1,2\}}_\textrm{\textit{k}-times}\}$ by performing a sequence of $3^k-1$ jumps of standard length, since the most accurate answer to the original question actually depends on which mathematical assumptions we are making at the beginning of the game when we decide to extend a planar chess piece to the third dimension and above. Our counterintuitive outcome follows from the observation that we can alternatively define a 2D knight as a piece that moves from one square to another on the chessboard by covering a fixed Euclidean distance of $\sqrt{5}$ so that also the statement of Theorem 3 in [Erde, J., Golénia, B., & Golénia, S. (2012), The closed knight tour problem in higher dimensions, The Electronic Journal of Combinatorics, 19(4), #P9] does not hold anymore for such a Euclidean knight, as long as a 2 × 2 × ⋯ × 2 chessboard with at least 2<sup>7</sup> cells is given. Moreover, we construct a classical closed knight's tour on $C(3,4)-\{(1,1,1,1)\}$ whose arrival is at a distance of 2 from $(1,1,1,1)$, and then we show a closed Euclidean knight's tour on $\{\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\}\subseteq \mathbb{Z}^7$.

Image Encryption Technology Based on Fractal Image Compression Algorithm

As the most commonly used information transmission method, digital images often store a large amount of personal information. To prevent information leakage, encrypting images is essential. Common image encryption techniques suffer from certain limitations, such as overly simple encryption methods and long encryption times. In response to the above issues, this study proposes the Frobenius canonical form image encryption scheme. It calculates the fractal code through the fractal compression algorithm and to encrypt the image, it adjusts the brightness coefficient in the fractal code. To address unsatisfactory correlation coefficients in encrypted images, the Frobenius canonical form image encryption is improved by introducing the Arnold transformation encryption, which combines the two methods to reduce correlation coefficients. Finally, the knight tour algorithm is put forward. In response to the long image scrambling time in the knight tour algorithm, the Tetragonal theorem is combined with the scheme to encrypt the image. It is then re-encrypted using the Frobenius canonical form. The experimental findings illustrate that when using Frobenius canonical form, Arnold transformation combined with Frobenius canonical form, and the tetragonal algorithm combined with knight tour algorithm to encrypt Lena images, the three decryption methods correspond to image similarity of over 70%, over 80%, and over 90%, respectively. Combining the tetragonal algorithm and the knight tour algorithm can significantly increase the security of image encryption.

A Modified KLEIN Encryption-based Knight Tour for Image Encryption

The security considerations should be balanced with the specific use case and potential risks associated with using lightweight encryption. The security offered by lighter encryption algorithms could not be as high as that offered by heavier encryption techniques. In this paper, a Modified KLEIN Algorithm is proposed for image encryption based on the Knight Tour movement in Chessboard. The required key generation is represented by inputting an image as a key image and then applying a specific operation based on knight tour movement to produce a key scheduled in the proposed encryption algorithm. The movement of Knight Tour applied in modifying the proposed algorithm for increasing security. The experimental results explain the efficiency of a modified algorithm when comparing the histogram of the input image with the encrypted image also the correlation is tested before and after encryption in three directions horizontal, vertical, and diagonal which explains there are very low values of them in all directions. The similarity test also explains there are high differences between the plain and encrypted images. The chessboard movement might be useful when used with another encryption algorithm which increases the confidentiality of transferring data. The contribution of this work is the use of an image as a key for encryption with a specific planning method which helps in key management.

High imperceptibility and robustness watermarking scheme for brain MRI using Slantlet transform coupled with enhanced knight tour algorithm

This research introduces a novel and robust watermarking scheme for medical Brain MRI DICOM images, addressing the challenge of maintaining high imperceptibility and robustness simultaneously. The scheme ensures privacy control, content authentication, and protection against the detachment of vital Electronic Patient Record information. To enhance imperceptibility, a Dynamic Visibility Threshold parameter leveraging the Human Visual System is introduced. Embeddable Zones and Non-Embeddable Zones are defined to enhance robustness, and an enhanced Knight Tour algorithm based on Slantlet Transform shuffles the embedding sequence for added security. The scheme achieves remarkable results with a Peak Signal-to-Noise Ratio (PSNR) evaluation surpassing contemporary techniques. Extensive experimentation demonstrates resilience to various attacks, with low Bit Error Rate (BER) and high Normalized Cross-Correlation (NCC) values. The proposed technique outperforms existing methods, emphasizing its superior performance and effectiveness in medical image watermarking.

A Blind Image Steganography Algorithm Based on Knight Tour Algorithm and QR Codes

Internet proliferation and technological progress have made multimedia information quickly accessible, but they have also posed a threat to privacy and security. Researchers have been interested in digital images due to their capacity to store large amounts of data due to the possibility of protecting sensitive information through digital steganography. Despite their visual imperceptibility, robustness, and ability to embed information, existing image steganography techniques face several challenges. To overcome these challenges, a novel image steganography approach based on a blind model strategy has been proposed for hiding covert messages. The model consists of two stages: embedding and extracting. In the embedding stage, a suitable cover image is selected using the FAST feature point detector. A text message is then converted to a QR code and embedded in the feature points' neighbors using knight tour steps in a chess game. The result is a stego image that appears identical to the original cover image but contains the secret message in its feature points' neighbors. The extracting stage involves finding the feature points and extracting the QR code to obtain the original text message. Since the feature points were not altered during the embedding process, the proposed model is known as a blind model. This approach eliminates the need for the original cover image during the extracting stage. The proposed model was evaluated using several metrics, including the peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM). The results demonstrate that the proposed algorithm can effectively embed and extract secret messages with high accuracy while maintaining the visual quality of the cover image with 100% of (SSIM), and 73.48 as an average of (PSNR).

Color-Gray Multi-Image Hybrid Compression-Encryption Scheme Based on BP Neural Network and Knight Tour.

In the research of multi-image encryption (MIE), the image type and size are important factors that limit the algorithm design. For this reason, the multi-image (MI) hybrid encryption algorithm that can flexibly encrypt color images and grayscale images of various sizes is proposed. Based on this, combining the back propagation (BP) neural network compression technology and the MI hybrid encryption algorithm, an MI hybrid compression-encryption (MIHCE) scheme can be obtained to reduce the pressure of simultaneous transmission and storage of multiple cipher images. Besides, two chaotic maps are used in the scheme design process. By plotting the phase diagrams under different parameter conditions, the rich variation of the behavior of the chaotic maps in the phase space is exhibited. The MIHCE scheme based on the chaotic maps consists of three parts: 1) compressing the MI cube by using the BP neural network; 2) scrambling the compressed MI cube based on the knight tour problem and chaotic sequences; and 3) diffusing the scrambled MI cube. After the MIHCE is completed, the obtained cipher images are stored and transmitted. Subsequently, the security analysis and compression performance analysis prove the feasibility and safety of the designed compression-encryption scheme.

Use of Several Grids in the Knight Tour’s Game to Find Shorter or Longer Biochemistry’s Sentences

Knight tour is a mathematical problem that was first solved by Leonhard Euler (1707-1785). The problem consists in finding if it is possible that the knight piece of chess can tour through all the boxes of a chess grid, passing only once through each box. Bishop piece can only move to diagonal boxes of one color, and pawns can only move in one column. It can be easily seen that king, queen and rook can move through all the boxes. But it was not clear that the knight could move all through. Euler found several solutions for the 8x8 grid, and he numbered all the boxes of the grid in the order through which the knight passed. Surprisingly, he obtained some solutions with semi-magic squares, in which the sum of all the numbers of each row and column was the same. Nevertheless, diagonals didn’t sum the same, as happens in the magic squares. One of the typical games in word game’s books is the knight tour, with a 5x5 grid, that hides a 25-syllable sentence. The objective of this work was to study other grids, to obtain different sentences length with 3x3 (9-syllables), 4x4 (16-syllables), 5x5 (25 syllables), 6x6 (36 syllables) or 7x7 (49 syllables) grids. Thus, this game would be more versatile and could be used more extensively than limiting it to 25-syllable sentences.

A new physics-based solar reconfigured model to enhance efficiency under partial shading conditions: Experimental feasibility

ABSTRACT Maximum power generation from solar modules is adversely affected by partial shading conditions. To strengthen and obtain feasible output under any complex shadings, a new double diode model with deterioration factor (DDM-DF) is proposed for the optimal selection of shading patterns. It is used to validate the effectiveness of the novel heuristic approach called time-distance-constrained topology (D-CT). The novel D-CT is compared with conventional techniques (bridge link (BL), series-parallel (S-P), total cross-tied (TCT)), conventional puzzles (sudoku (SDK), magic-square (M-SP)), and recently reconfigured techniques (permutation-combination (PCR), knight-tour (KTR)). In this paper, the experimental study validates the outcomes obtained by MATLAB/SIMULINK results under various realistic shadings. Experimental results prove that the novel proposed D-CT enhances the efficiency by 5.45%, 3.57%, and 3.54% over TCT, PCR, and KTR under inverted-L shading. Further, the minimum and maximum mismatch loss reduced by the proposed D-CT is up to 1.90% and 97.61% under top shading against SDK and S-P respectively. The real-time validation for the proposed novel 4×4 D-CT under pi and diagonal shading exhibits greater performance parameter enhancement of a solar array subjected to shadings in correlation with other models.

Modelling and Analysis of Hybrid Transformation for Lossless Big Medical Image Compression

Due to rapidly developing technology and new research innovations, privacy and data preservation are paramount, especially in the healthcare industry. At the same time, the storage of large volumes of data in medical records should be minimized. Recently, several types of research on lossless medically significant data compression and various steganography methods have been conducted. This research develops a hybrid approach with advanced steganography, wavelet transform (WT), and lossless compression to ensure privacy and storage. This research focuses on preserving patient data through enhanced security and optimized storage of large data images that allow a pharmacologist to store twice as much information in the same storage space in an extensive data repository. Safe storage, fast image service, and minimum computing power are the main objectives of this research. This work uses a fast and smooth knight tour (KT) algorithm to embed patient data into medical images and a discrete WT (DWT) to protect shield images. In addition, lossless packet compression is used to minimize memory footprints and maximize memory efficiency. JPEG formats' compression ratio percentages are slightly higher than those of PNG formats. When image size increases, that is, for high-resolution images, the compression ratio lies between 7% and 7.5%, and the compression percentage lies between 30% and 37%. The proposed model increases the expected compression ratio and percentage compared to other models. The average compression ratio lies between 7.8% and 8.6%, and the expected compression ratio lies between 35% and 60%. Compared to state-of-the-art methods, this research results in greater data security without compromising image quality. Reducing images makes them easier to process and allows many images to be saved in archives.

A Robust Audio Steganography Method Using Partial Knight Tour for Concealing Messages of Text and Image

The rapid growth in transfer of data over the Internet realized the need to secured data transmission. Moreover, there is a need to send message that contains a text and image. The steganography is the most promising technique for facilitating the secure communication without affecting the data quality. In this paper, we propose a robust audio steganography approach that hides secret messages of text and image in an audio WAV file using partial knight tour positions in 4×8 square blocks. The cover audio is divided into 4×8 (or 32) audio sample blocks. The secret data are converted to bits and selected 4×8 audio bytes then can hide ASCII codes of 4 characters. The hiding of secret data block is based on a random selected starting audio byte, and on a random selected tour of 32 predefined tours. The algorithm changes the least significant bit (LSB) of the audio bytes based, as mentioned, on randomly selected tours and randomly selected starting audio byte. The resulted stego file can be, then, transmitted over the communication channel such as Internet to the receiver side. After receiving, the extraction process is performed to extract the secret message. The proposed algorithm is implemented us[ing C# programming languages. The PSNR and MSE parameters are used to evaluate the efficiency of the proposed algorithm and they will be checked after implementation. The proposed technique provides a high quality of Peak Signal-to-Noise Ratio (PSNR) and Signal to Noise Ratio (SNR).

Photovoltaic array reconfiguration under partial shading conditions for maximum power extraction via knight's tour technique

AbstractThis paper introduces a novel reconfiguration technique, called Knight's tour to extract maximum power from photovoltaic (PV) arrays in partial shading conditions. The Knight's tour reconfigures the PV arrays based on the Knight's movements on the chessboard. The proposed procedure achieves the maximum power values by spreading partial shadows in all rows. Knight's tour can be applied to a variety of PV arrays in different dimensions and sizes. Accordingly, the Knight's tour procedure is applied to four cases in square and rectangular shapes with different dimensions and various shading conditions in each case. To make a direct comparison and present the effectiveness of the suggested procedure, the total-cross-tied connection model and conventional methods such as SuDoKu, optimal SuDoKu, improved SuDoKu, and Skyscraper puzzle are also implemented to the introduced cases. The results of the maximum power point tracking in each case are evaluated by indicators such as global maximum power point (GMPP), fill factor, mismatch loss, and efficiency. Finally, evaluations emphasize the ability and effectiveness of the Knight's tour solution compared to other methods by achieving the GMPP values such as $$74.7\,{I}_{m}{V}_{m}$$ 74.7 I m V m , $$66.6\,{I}_{m}{V}_{m}$$ 66.6 I m V m , $$46.8\,{I}_{m}{V}_{m}$$ 46.8 I m V m , and $$109.8\,{I}_{m}{V}_{m}$$ 109.8 I m V m for cases 1 to 4, respectively. The Knight's tour method can be utilized as an efficient tool for the PV arrays in real-world systems that suffer from partial shading.

Taming the knight's tour: Minimizing turns and crossings

We introduce two new metrics of “simplicity” for knight's tours: the number of turns and the number of crossings. We give a novel algorithm that produces tours with 9.25n+O(1) turns and 12n+O(1) crossings on an n×n board, and we show lower bounds of (6−ϵ)n and 4n−O(1) on the respective problems of minimizing these metrics. Hence, our algorithm achieves approximation ratios of 9.25/6+o(1) and 3+o(1). Our algorithm takes linear time and is fully parallelizable, i.e., the tour can be computed in O(n2/p) time using p processors in the CREW PRAM model. We generalize our techniques to rectangular boards, high-dimensional boards, symmetric tours, odd boards with a missing corner, and tours for (1,4)-leapers. In doing so, we show that these extensions also admit a constant approximation ratio on the minimum number of turns, and on the number of crossings in most cases.

An optimized knight traversal technique to detect multiple faults and Module Sequence Graph based reconfiguration of microfluidic biochip

Conventional biomedical analysers are replaced by digital microfluidic biochips and they are adequate to integrate different biomedical functions, essential for diverse bioassay operations. From the last decade, microfluidic biochips are getting plenty of acceptances in the field of miscellaneous healthcare sectors like DNA analysis, drug discovery and clinical diagnosis. These devices are also bearing a vital role in the area of safety critical applications such as food safety testing, air quality monitoring etc. As these devices are used in safety critical applications, clinical diagnosis and real-time biomolecular assay operations, these must have properties like precision, reliability and robustness. To accept it for discriminating purposes, the microfluidic device must endorse its preciseness and strength by following sublime testing strategy. Here, an optimized droplet traversal technique is proposed to investigate the multiple defective electrodes of a digital microfluidic biochip by embedding boundary cum row traversal and KNIGHT traversal procedure (based on the famous Knight Tour Problem). The proposed approach also enumerates the traversal time for a fault-free biochip. In addition to identifying the faulty electrodes, a Module Sequencing Graph based reconfiguration technique is proposed here to reinstate the device for normal bioassay operation.

Comparative Study on Image Steganography: Knight Tour and Rivest Cipher Four Algorithms

Image steganography is a process of hiding message behind an image file which focuses on protecting the existence of a message secret. There is a security risk in the current image steganography process. Since stego-image will be transferred on unsecured Internet network, attackers will attack and try to decode the message behind the stego-image because of the vulnerable algorithm. Therefore, it is very important to search for a method to make the process of encoding the stego-image more secure. There are many algorithms developed to make the stego-image become more secured. However, the usage of Knight Tour (KT) and Rivest Cipher Four (RC4) algorithms in image steganography are still insufficient although that the algorithms are claimed to be secured and robust. KT algorithm is an easy mathematical technique that can increase the security of hidden information, meanwhile, RC4 is known as a simple algorithm but systematic in cover image programming. In this paper, the performance of KT and RC4 algorithms are observed to measure the security and robustness of JPG image format. Peak Signal to Noise Ratio (PSNR) and Mean Square Error (MSE) are used to observe the image quality to improve the security factor in the stego-image. From the results, it is found that KT generated better performance compared to RC4.

The use of Least Significant Bit (LSB) and Knight Tour Algorithm for image steganography of cover image

<span lang="EN-US">Steganography is one of the method to communicate in a hidden way. In another word, steganography literally means the practice of hiding messages or information within another data. Previous studies have proposed various steganography techniques using different approaches including Least Significant Bit (LSB), Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT). However, different approaches still have its own weaknesses. Therefore image stenography using Knight Tour Algorithm with Least Significant Bit (LSB) technique is presented. The main objective is to improve the security factor in the stego image. Basically, the proposed technique is divided into two parts which are the sender and receiver side. Then, steganalysis which is a type of attack on stenography algorithm is used to detect the secret message in the cover image by the statistical analysis of pixel values. Chi Square Statistical Attach which is one of the type of steganalysis is used to detect these near-equal Po Vs in images and bases the probability of embedding on how close to equal the even pixel values and their corresponding odd pixel values are in the test image. The Knight Tour Algorithm is applied due to the common Least Significant Bit technique that is weak in security and easily decoded by outsider.</span>

Knight Tour for Image Steganography Technique

The growth rate of the Internet is exceeding that of any previous technology. As the Internet has become the major medium for transferring sensitive information, the security of the transferred message has now become the utmost priority. To ensure the security of the transmitted data, Image steganography has emerged out as an eminent tool of information hiding. The frequency of availability of image file is high and provides high capacity. In this paper, a method of secure data hiding in image is proposed that uses knight tour positions and further 8-queen positions in 8*8 pixel blocks.The cover image is divided into 8*8 pixel blocks and pixels are selected from each block corresponding to the positions of Knight in 8*8 chessboard starting from different pixel positions. 8-pixel values are selected from alternate knight position. Selected pixels values converted to 8-bit ASCII code and result in 8* 8 bit matrix. 8-Queen’s solution on 8*8 chessboard is applied on 8*8 bit matrix. The bits selected from 8-Queens positions and compared with 8-bit ASCII code of message characters. The proposed algorithm changes the LSB of only some of the pixels based on the above comparison. Based on parameters like PSNR and MSE the efficiency of the method is checked after implementation. Then the comparison done with some already proposed techniques. This is how, image steganography showed interesting and promising results when compared with other techniques.