Mersenne Numbers in the Bases of Systems of Residual Classes when Transmitting Data in Serial Communication Channels

Смирнов, Александр Александрович; Bondar, V. V.; Rozhenko, O.D.; Mirzoyan, Marine; Darjania, A. D.

doi:10.1007/s10958-022-05688-0

Cited by 2 publications

(1 citation statement)

References 2 publications

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“…There are works in which the expediency of using such numbers for data transmission is justified [21,22], etc.…”

Section: Introductionmentioning

confidence: 99%

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

Suleimenov,

Vitulyova,

Matrassulova

2023

PLoS ONE

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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 algorithm allows, among other things, to reduce the multiplication operation modulo a prime number p = 2n+1 to an addition operation.

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“…There are works in which the expediency of using such numbers for data transmission is justified [21,22], etc.…”

Section: Introductionmentioning

confidence: 99%

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

Suleimenov,

Vitulyova,

Matrassulova

2023

PLoS ONE

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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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Mersenne Numbers in the Bases of Systems of Residual Classes when Transmitting Data in Serial Communication Channels

Cited by 2 publications

References 2 publications

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

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

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

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