Search Results (45)

The S-box is a key component of modern ciphers, determining the quality and performance of the cryptographic algorithms in which it is applied. Many constructions for synthesizing high-quality S-boxes have been established, and those based on Galois fields theory—for example, the Nyberg construction applied in the AES cryptographic algorithm—are particularly important. An integral component of the Nyberg construction is the affine transformation, which is used to improve the avalanche and correlation properties of the S-box. In this paper, a new approach is adopted for synthesizing affine transformations for S-boxes based on the quaternary matrices over the Galois field GF(4). We describe four basic structures that serve as the foundation for synthesizing a complete class of 648 affine transformation matrices of order n = 3 and a class of 7776 matrices of order n = 4 and introduce a recurrent structure to facilitate the synthesis of matrices for higher orders. Using these matrices in combination with the Nyberg construction, it is possible to construct bijective S-boxes that outperform the original Nyberg construction and many other known S-boxes in terms of strict avalanche criterion (SAC) and bit independence criterion strict avalanche criterion (BIC SAC) values, while maintaining a maximal level of nonlinearity and good cryptographic properties. We also propose modified GF(4) affine transformations that can be applied to specialized S-boxes which already satisfy the SAC for both component Boolean and 4-functions, as well as the criterion of minimal correlation between input and output, allowing us to enhance their nonlinearity to the value of N_f = 96. We integrate the synthesized S-boxes into the AES algorithm and evaluate their practical performance. The encryption outputs successfully pass the NIST statistical test suite in 96 out of 100 cases, outperforming both the original AES S-box and other reference constructions, confirming the practical strength of the proposed method. Full article

Substitution boxes (S-Boxes) function as essential nonlinear elements in contemporary cryptographic systems, offering robust protection against cryptanalytic attacks. This study presents a novel technique for generating compact 8-bit S-Boxes based on multiplication in the Galois Field $GF\left(2^4\right)$. The goal of this method is to create S-Boxes with low hardware implementation cost while ensuring cryptographic properties. Experimental results indicate that the suggested S-Boxes achieve a nonlinearity value of 112, matching the AES S-Box. They also maintain other cryptographic properties, such as the Bit Independence Criterion (BIC), the Strict Avalanche Criterion (SAC), Differential Approximation Probability, and Linear Approximation Probability, within acceptable security thresholds. Notably, compared to existing studies, the proposed S-Box architecture demonstrates enhanced hardware efficiency, significantly reducing resource utilization in implementations. Specifically, the implementation cost of the S-Box consists of 31 XOR gates, 32 two-input AND gates, 6 two-input OR gates, and 2 MUX21s. Moreover, this work provides a thorough assessment of the S-Box, covering cryptographic properties, side channel attacks, and implementation aspects. Furthermore, the study estimates the quantum resource requirements for implementing the S-Box, including an analysis of CNOT, Toffoli, and NOT gate counts. Full article

In this paper, we discuss non-binary polar codes using a $2\times 2$ matrix over a Galois field GF($2^q$) as the kernel. Conventional construction of non-binary polar codes divides the synthesized channels into frozen channels and information channels. Each information channel carries one symbol, i.e., q bits. However, there are many middle channels with insufficient polarization, which cannot carry one symbol of q bits but only i bits, $1\le i<q,i\in \mathbb{Z}$, at finite block length. In this paper, we consider bit-level construction for multiplicative repetition (MR)-based non-binary polar codes and propose a bit-level construction based on the two following methods. We first calculate the error probability and channel capacity lower bound of each synthesized channel based on the channel degradation method, and then determine both the number and index of the carried bits for each synthesized channel according to the symbol error probability and capacity. To reduce complexity, we also introduce a Monte-Carlo method. We compute the error probability of each synthesized channel carrying i information bits and select the optimal construction that can minimize the union bound of the error probability. Finally, an improved construction-based probabilistic shaping method for MR-based non-binary polar codes is considered. Simulation results show that the proposed construction significantly improved the decoding performance compared with the conventional construction scheme. Full article

The method of non-standard algebraic extensions based on the use of additional formal solutions of the reduced equations is extended to the case corresponding to three-dimensional space. This method differs from the classical one in that it leads to the formation of algebraic rings rather than fields. The proposed approach allows one to construct a discrete coordinate system in which the role of three basis vectors is played by idempotent elements of the ring obtained by a non-standard algebraic extension. This approach allows, among other things, the identification of the symmetry properties of objects defined through discrete Cartesian coordinates, which is important, for example, when using advanced methods of digital image processing. An explicit form of solutions of the equations is established that allow one to construct idempotent elements for Galois fields $GF\left(p\right)$ such that $p-1$ is divisible by three. The possibilities of practical use of the proposed approach are considered; in particular, it is shown that the use of discrete Cartesian coordinates mapped onto algebraic rings is of interest from the point of view of improving UAV swarm control algorithms. Full article

The Internet of Things (IoT), which is characteristic of the current industrial revolutions, is the connection of physical devices through different protocols and sensors to share information. Even though the IoT provides revolutionary opportunities, its connection to the current Internet for smart cities brings new opportunities for security threats, especially with the appearance of new threats like quantum computing. Current approaches to protect IoT data are not immune to quantum attacks and are not designed to offer the best data management for smart city applications. Thus, post-quantum cryptography (PQC), which is still in its research stage, aims to solve these problems. To this end, this research introduces the Dynamic Galois Reed–Solomon with Quantum Resilience (DGRS-QR) system to improve the secure management and communication of data in IoT smart cities. The data preprocessing includes K-Nearest Neighbors (KNN) and min–max normalization and then applying the Galois Field Adaptive Expansion (GFAE). Optimization of the quantum-resistant keys is accomplished by applying Artificial Bee Colony (ABC) and Moth Flame Optimization (MFO) algorithms. Also, role-based access control provides strong cloud data security, and quantum resistance is maintained by refreshing keys every five minutes of the active session. For error correction, Reed–Solomon (RS) codes are used which provide data reliability. Data management is performed using an attention-based Bidirectional Long Short-Term Memory (Att-Bi-LSTM) model with skip connections to provide optimized city management. The proposed approach was evaluated using key performance metrics: a key generation time of 2.34 s, encryption time of 4.56 s, decryption time of 3.56 s, PSNR of 33 dB, and SSIM of 0.99. The results show that the proposed system is capable of protecting IoT data from quantum threats while also ensuring optimal data management and processing. Full article

It is shown that it is reasonable to use Galois fields, including those obtained by algebraic extensions, to describe the position of a point in a discrete Cartesian coordinate system in many cases. This approach is applicable to any problem in which the number of elements (e.g., pixels) into which the considered fragment of the plane is dissected is finite. In particular, it is obviously applicable to the processing of the vast majority of digital images actually encountered in practice. The representation of coordinates using Galois fields of the form GF(p²) is a discrete analog of the representation of coordinates in the plane through a complex variable. It is shown that two different types of algebraic extensions can be used simultaneously to represent transformations of discrete Cartesian coordinates described through Galois fields. One corresponds to the classical scheme, which uses irreducible algebraic equations. The second type proposed in this report involves the use of a formal additional solution of some equation, which has a usual solution. The correctness of this approach is justified through the representation of the elements obtained by the algebraic expansion of the second type by matrices defined over the basic Galois field. It is shown that the proposed approach is the basis for the development of new methods of information protection, designed to control groups of UAVs in the zone of direct radio visibility. The algebraic basis of such methods is the solution of systems of equations written in terms of finite algebraic structures. Full article

Statistical analysis shows that the most common errors in the transmission of information consist of single errors and transposition errors. Error detection and correction methods are often desired, particularly when the accuracy of information is of crucial importance. Inspired by a check digit system constructed from the companion matrix of a primitive polynomial over the integers ${\mathbb{Z}}_p$ and that focused on error detection, this work develops error-correction formulas for single errors and transposition errors for that check digit scheme. We also propose an application to DNA sequences. Full article

An RSA generalization using complex integers was introduced by Elkamchouchi, Elshenawy and Shaban in 2002. This scheme was further extended by Cotan and Teșeleanu to Galois fields of order $n\ge 1$. In this generalized framework, the key equation is $ed-k(p^n-1)(q^n-1)=1$, where p and q are prime numbers. Note that the classical RSA and Elkamchouchi et al.’s key equations are special cases, namely, when $n=1$ and $n=2$. In addition to introducing this generic family, Cotan and Teșeleanu described a continued fractions attack capable of recovering the secret key d if $d<N^{0.25n}$. This bound was later improved by Teșeleanu using a lattice-based method. In this paper, we explore other lattice attacks that could lead to factoring the modulus $N=pq$, namely, we propose a series of partial exposure attacks that can aid an adversary in breaking this family of cryptosystems if certain conditions hold. Full article

Mathematical tools have been developed that are analogous to the tool that allows one to reduce the description of linear systems in terms of convolution operations to a description in terms of amplitude-frequency characteristics. These tools are intended for use in cases where the system under consideration is described by partial digital convolutions. The basis of the proposed approach is the Fourier–Galois transform using orthogonal bases in corresponding fields. As applied to partial convolutions, the Fourier–Galois transform is decomposed into a set of such transforms, each of which corresponds to operations in a certain Galois field. It is shown that for adequate application of the Fourier–Galois transform to systems described by partial convolutions, it is necessary to ensure the same number of cycles in each of the transforms from the set specified above. To solve this problem, the method of algebraic extensions was used, a special case of which is the transition from real numbers to complex numbers. In this case, the number of cycles varies from $p$ to $p^n/k$, where $p$ is a prime number, $n$ and $k$ are integers, and an arbitrary number divisor of $p^n$ can be chosen as $k$. This allows us to produce partial Fourier–Galois transforms corresponding to different Galois fields, for the same number of cycles. A specific example is presented demonstrating the constructiveness of the proposed approach. Full article

The proliferation of Internet of Things (IoT) devices has introduced significant security challenges, including weak authentication, insufficient data protection, and firmware vulnerabilities. To address these issues, we propose a linguistic secret sharing scheme tailored for IoT applications. This scheme leverages neural networks to embed private data within texts transmitted by IoT devices, using an ambiguous token selection algorithm that maintains the textual integrity of the cover messages. Our approach eliminates the need to share additional information for accurate data extraction while also enhancing security through a secret sharing mechanism. Experimental results demonstrate that the proposed scheme achieves approximately 50% accuracy in detecting steganographic text across two steganalysis networks. Additionally, the generated steganographic text preserves the semantic information of the cover text, evidenced by a BERT score of 0.948. This indicates that the proposed scheme performs well in terms of security. Full article

A scheme for the Fourier–Galois spectrum analyzer for the field GF(31) is proposed. It is shown that this analyzer allows for solving a wide enough range of problems related to image processing, in particular those arising in the course of experimental studies in the field of physical chemistry. Such images allow digital processing when divided into a relatively small number of pixels, which creates an opportunity to use Galois fields of relatively small size. The choice of field GF(31) is due to the fact that the number 31 is a Mersenne prime number, which considerably simplifies the algorithm of calculating the Fourier–Galois transform in this field. The proposed scheme of the spectrum analyzer is focused on the use of threshold sensors, at the output of which signals corresponding to binary logic are formed. Due to this fact, further simplification of the proposed analyzer scheme is achieved. The constructiveness of the proposed approach is proven using digital modeling of electronic circuits. It is concluded that when solving applied problems in which an image can be divided into a relatively small number of pixels, it is important to take into account the specificity of particular Galois fields used for their digital processing. Full article

Data security and privacy have become essential due to the increasingly advanced interconnectivity in today’s world, hence the reliance on cryptography. This paper introduces a new algorithm that uses a novel hybrid Tent–May chaotic map to generate pseudo-random numbers, as well as block encryption. We design a robust S-box by combining the Tent and May Maps, which yields a chaotic system with improved cryptographic properties. This S-box is a critical cryptographic primitive that significantly improves encryption security and leverages the strengths of both maps. The encryption process involves two key steps: block-wise substitution and permutation. First, we divide the image into $16\times 16$ blocks, then substitute each pixel with the $8-byte$ key and S-box. Next, we convert the encrypted image back into vector form, reorganize it using the permutation vector based on the subgroups of $S_{16}$, and finally return it to its original form. This approach greatly improves block cipher security when used, especially to protect medical images by guaranteeing their confidentiality and noninterference. Performance measures like PSNR, UACI, MSE, NCC, AD, SC, MD, and NAE prove how immune our method is to various cryptographic and statistical attacks, making it more accurate and more secure than the existing techniques. Full article

Data security is one of the biggest concerns in the modern world due to advancements in technology, and cryptography ensures that the privacy, integrity, and authenticity of such information are safeguarded in today’s digitally connected world. In this article, we introduce a new technique for the construction of non-linear components in block ciphers. The proposed S-box generation process is a transformational procedure through which the elements of a finite field are mapped onto highly nonlinear permutations. This transformation is achieved through a series of algebraic and combinatorial operations. It involves group actions on some pairs of two Galois fields to create an initial S-box $P_{r}Sbox$, which induces a rich algebraic structure. The post S-box $P_{o}Sbox$, which is derived from heuristic group-based optimization, leads to high nonlinearity and other important cryptographic parameters. The proposed S-box demonstrates resilience against various attacks, making the system resistant to statistical vulnerabilities. The investigation reveals remarkable attributes, including a nonlinearity score of 112, an average Strict Avalanche Criterion score of 0.504, and LAP (Linear Approximation Probability) score of 0.062, surpassing well-established S-boxes that exhibit desired cryptographic properties. This novel methodology suggests an encouraging approach for enhancing the security framework of block ciphers. In addition, we also proposed a three-step image encryption technique comprising of Row Permutation, Bitwise XOR, and block-wise substitution using $P_{o}Sbox$. These operations contribute to adding more levels of randomness, which improves the dispersion across the cipher image and makes it equally intense. Therefore, we were able to establish that the approach works to mitigate against statistical and cryptanalytic attacks. The PSNR, UACI, MSE, NCC, AD, SC, MD, and NAE data comparisons with existing methods are also provided to prove the efficiency of the encryption algorithm. Full article

A method is proposed that reduces the computation of the reduced digital convolution operation to a set of independent convolutions computed in Galois fields. The reduced digital convolution is understood as a modified convolution operation whose result is a function taking discrete values in the same discrete scale as the original functions. The method is based on the use of partial convolutions, reduced to computing a modulo integer $q_0$, which is the product of several prime numbers: $q_0=p_1{p}_2\dots p_n$. It is shown that it is appropriate to use the expansion of the number $q_0$, to $q=p_0{p}_1{p}_2\dots p_n$, where $p_0$ is an additional prime number, to compute the reduced digital convolution. This corresponds to the use of additional digits in the number system used to convert to partial convolutions. The inverse procedure, i.e., reducing the result of calculations modulo q to the result corresponding to calculations modulo $q_0$, is provided by the formula that used only integers proved in this paper. The possibilities of practical application of the obtained results are discussed. Full article

Secret sharing is a data security technique that divides secret information into multiple parts, embeds these parts into various shares, and distributes these shares to different participants. The original secret information can be retrieved only when the number of shares gathered meets a required threshold. This paper proposes a secret sharing method that can hide data in encrypted images with reversibility and allows content owners to add an additional layer of security before uploading data to the cloud. This method enables the independent extraction of images and data, ensuring that the recovered images and extracted data can serve as validation information for each other. The proposed method not only enhances data security but also guarantees the accuracy of the extracted information. Full article