Features of digital signal processing algorithms using Galois fields GF(2n+1)

Suleimenov, Ibragim E.; Vitulyova, Yelizaveta S.; Matrassulova, Dinara K.

doi:10.1371/journal.pone.0293294

PLoS ONE

2023

DOI: 10.1371/journal.pone.0293294

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Features of digital signal processing algorithms using Galois fields GF(2n+1)

Ibragim E. Suleimenov,

Yelizaveta S. Vitulyova,

Dinara K. Matrassulova

Abstract: An alternating representation of integers in binary form is proposed, in which the numbers -1 and +1 are used instead of zeros and ones. It is shown that such a representation creates considerable convenience for multiplication numbers modulo p = 2n+1. For such numbers, it is possible to implement a multiplication algorithm modulo p, similar to the multiplication algorithm modulo the Mersenne number. It is shown that for such numbers a simple algorithm for digital logarithm calculations may be proposed. This a… Show more

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2024

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Cited by 3 publications

References 23 publications

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The specifics of the Galois field GF(257) and its use for digital signal processing

Bakirov,

Matrassulova,

Vitulyova

et al. 2024

Sci Rep

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An algorithm of digital logarithm calculation for the Galois field $$GF(257)$$ G F ( 257 ) is proposed. It is shown that this field is coupled with one of the most important existing standards that uses a digital representation of the signal through 256 levels. It is shown that for this case it is advisable to use the specifics of quasi-Mersenne prime numbers, representable in the form $${p=2}^{n}+1$$ p = 2 n + 1 , which includes the number 257. For fields $$GF({2}^{n}+1)$$ G F ( 2 n + 1 ) , an alternating encoding can be used, in which non-zero elements of the field are displayed through binary characters corresponding to the numbers + 1 and − 1. In such an encoding, multiplying a field element by 2 is reduced to a quasi-cyclic permutation of binary symbols (the permuted symbol changes sign). Proposed approach makes it possible to significantly simplify the design of computing devices for calculation of digital logarithm and multiplication of numbers modulo 257. A concrete scheme of a device for digital logarithm calculation in this field is presented. It is also shown that this circuit can be equipped with a universal adder modulo an arbitrary number, which makes it possible to implement any operations in the field under consideration. It is shown that proposed digital algorithm can also be used to reduce 256-valued logic operations to algebraic form. It is shown that the proposed approach is of significant interest for the development of UAV on-board computers operating as part of a group.

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The specifics of the Galois field GF(257) and its use for digital signal processing

Bakirov,

Matrassulova,

Vitulyova

et al. 2024

Sci Rep

View full text Add to dashboard Cite

show abstract

Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions

Kadyrzhan,

Bakirov,

Shaltykova

et al. 2024

Algorithms

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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 pn/k, where p is a prime number, n and k are integers, and an arbitrary number divisor of pn 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.

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Peculiarities of Applying Partial Convolutions to the Computation of Reduced Numerical Convolutions

Suleimenov,

Kadyrzhan,

Matrassulova

et al. 2024

Applied Sciences

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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 q0, which is the product of several prime numbers: q0=p1p2…pn. It is shown that it is appropriate to use the expansion of the number q0, to q=p0p1p2…pn, where p0 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 q0, is provided by the formula that used only integers proved in this paper. The possibilities of practical application of the obtained results are discussed.

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Features of digital signal processing algorithms using Galois fields GF(2n+1)

Cited by 3 publications

References 23 publications

The specifics of the Galois field GF(257) and its use for digital signal processing

The specifics of the Galois field GF(257) and its use for digital signal processing

Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions

Peculiarities of Applying Partial Convolutions to the Computation of Reduced Numerical Convolutions

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