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

An algorithm of digital logarithm calculation for the Galois field 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...

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Vydáno v:Scientific reports Ročník 14; číslo 1; s. 15376 - 15
Hlavní autoři: Bakirov, Akhat, Matrassulova, Dinara, Vitulyova, Yelizaveta, Shaltykova, Dina, Suleimenov, Ibragim
Médium: Journal Article
Jazyk:angličtina
Vydáno: London Nature Publishing Group UK 04.07.2024
Nature Publishing Group
Nature Portfolio
Témata:
639/705/1041
639/705/1042
639/705/258
Algorithms
Computers
Digital logarithm
Galois fields
Humanities and Social Sciences
Mersenne numbers
Modulo multiplication algorithm
multidisciplinary
Prime numbers
Quasi-cyclic permutations
Science
Science (multidisciplinary)
Signal processing
UAV flight computers
Quasi-cyclic permutations
Mersenne numbers
Modulo multiplication algorithm
UAV flight computers
Galois fields
Digital logarithm
ISSN:2045-2322, 2045-2322
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Shrnutí:An algorithm of digital logarithm calculation for the Galois field 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 , which includes the number 257. For fields 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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ISSN:2045-2322
2045-2322
DOI:10.1038/s41598-024-66332-2