Algebraic fields and rings as a digital signal processing tool

Matrassulova, Dinara K.; Vitulyova, Yelizaveta S.; Konshin, Sergey; Сулейменов, И. Э.

doi:10.11591/ijeecs.v29.i1.pp206-216

Cited by 6 publications

(7 citation statements)

References 24 publications

(49 reference statements)

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“…Finite fields, also known as Galois fields (GF), are mathematical structures that have important properties and applications in various fields, including cryptography, error correction codes, and digital signal processing [11] [12]. Finite fields play a crucial role in modern cryptography algorithms.…”

Section: An Overview Of Finite Fieldmentioning

confidence: 99%

“…[10] describes a brand-new digital watermarking method for color photographs that relies on the discrete cosine transform (DCT) and a triple-byte nonlinear block cipher. Further [11] proposed a new digital watermarking method for color images based on a triple-byte nonlinear block cipher and the discrete cosine transform (DCT). Based on the Galois ring (GR 23,8), a triplebyte nonlinear part of a block cipher, specifically a 24x24 substitution box (S-box), was created.…”

Section: Introductionmentioning

confidence: 99%

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Enhancing Speed and Imperceptibility in Watermarking Systems by Leveraging Galois Field Tables

Hassan,

Rahma

2024

IEEE Access

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The use of watermarking techniques, which incorporate invisible information into multimedia files, is crucial for securing digital content. Watermarking research still faces difficulties in striking a balance between speed, robustness, and imperceptibility. This research suggests a novel method for improving the speed and imperceptibility of watermarking systems that makes use of Galois Field (GF) tables. The advantage of creating GF tables is that they implement GF operations well, which greatly accelerates the watermarking processes. It is possible to reduce computational complexity and speed up execution times by previously calculating and storing GF tables. The proposed method is ideal for applications with strict time limitations since this performance improvement allows real-time watermarking. Additionally, imperceptibility is enhanced by the use of well-chosen irreducible polynomials in GF-based watermarking algorithms. The distribution and modulation of watermark bits are influenced by the selection of irreducible polynomials, allowing for optimal embedding with the least amount of visual distortions. By using this selection technique, the host media's quality and integrity are preserved because the embedded watermark is kept transparent or undetectable to human observers. In conclusion, this study provides a novel approach that makes use of Galois Field tables and selective irreducible polynomials to improve the speed and imperceptibility of watermarking systems. GF tables are used to increase computing effectiveness and enable real-time processing, whereas irreducible polynomials are selected to maximize imperceptibility. This strategy creates new opportunities for the creation of watermarking methods that effectively safeguard digital content without reducing user experience or system performance.

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Section: An Overview Of Finite Fieldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Enhancing Speed and Imperceptibility in Watermarking Systems by Leveraging Galois Field Tables

Hassan,

Rahma

2024

IEEE Access

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“…Like the classical analogue, this theorem allows you to reduce the convolution operation to the multiplication operation. The difference is that for digital signals corresponding to a certain set of discrete levels, the Fourier-Galois transform should be used, i.e., the signal model (as well as the kernel model of the convolution operator) is functions that take values in Galois fields or even finite algebraic rings [13].…”

Section: Introductionmentioning

confidence: 99%

Convolutional neural networks from the perspective of the problems of multivalued logic

Vitulyova,

Matrassulova,

SULEIMENOV

et al. 2023

Preprint

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It is shown that it is possible to reduce the operations performed by convolutional neural networks to algebraic, and then to operations of multivalued logic. The tool for this is signal models, which are functions that take values in Galois fields and algebraic rings. This approach, among other things, allows us to switch to the use of partial convolutions, which significantly simplify the analysis of digital signals and images. Such convolutions are performed directly in terms of Galois fields, which makes it possible to use a digital analogue of the classical convolution theorem. The advantage in this regard is the possibility of using a digital analogue of transfer functions, because in terms of Fourier-Galois spectra, the convolution operation is reduced to the multiplication operation by the transfer function. This, in turn, allows us to exhaustively describe a convolutional neural network through a set of transfer functions that take values in Galois fields. The close connection between Galois fields and multivalued logics makes it possible to describe the functioning of convolutional neural networks using variables of multivalued logic.

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“…The tool for generating the algebraic δ-function for the cases under consideration is the transition from the use of algebraic fields to finite algebraic rings, which are already widely used in information technology 30 , 31 .…”

Section: Introductionmentioning

confidence: 99%

“…We also note that the issues under consideration are also important from the point of view of improving the methods of digital signal and image processing. As shown in 30 , 35 , 36 , it is permissible to use functions that take values in Galois fields and/or algebraic rings to simulate digital signals. This allows you to move on to signal processing tools based on multivalued logic.…”

Section: Introductionmentioning

confidence: 99%

Improving the efficiency of using multivalued logic tools: application of algebraic rings

Suleimenov,

Vitulyova,

Kabdushev

et al. 2023

Sci Rep

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It is shown that in order to increase the efficiency of using methods of abstract algebra in modern information technologies, it is important to establish an explicit connection between operations corresponding to various varieties of multivalued logics and algebraic operations. For multivalued logics, the number of variables in which is equal to a prime number, such a connection is naturally established through explicit algebraic expressions in Galois fields. It is possible to define an algebraic δ-function, which allows you to reduce any truth table to an algebraic expression, for the case when the number of values accepted by a multivalued logic variable is equal to an integer power of a prime number. In this paper, we show that the algebraic δ-function can also be defined for the case when the number of values taken by a multivalued logic variable is p − 1, where p is a prime number. This function also allows to reduce logical operations to algebraic expressions. Specific examples of the constructiveness of the proposed approach are presented, as well as electronic circuits that experimentally prove its adequacy.

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Algebraic fields and rings as a digital signal processing tool

Cited by 6 publications

References 24 publications

Enhancing Speed and Imperceptibility in Watermarking Systems by Leveraging Galois Field Tables

Enhancing Speed and Imperceptibility in Watermarking Systems by Leveraging Galois Field Tables

Convolutional neural networks from the perspective of the problems of multivalued logic

Improving the efficiency of using multivalued logic tools: application of algebraic rings

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