Fast Kötter–Nielsen–Høholdt interpolation over skew polynomial rings and its application in coding theory
Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metr...
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| Vydané v: | Designs, codes, and cryptography Ročník 92; číslo 2; s. 435 - 465 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
New York
Springer US
01.02.2024
Springer Nature B.V |
| Predmet: | |
| ISSN: | 0925-1022, 1573-7586 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of Kötter–Nielsen–Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed–Solomon codes in the sum-rank metric and skew Reed–Solomon codes in the skew metric requiring at most
O
~
s
ω
M
(
n
)
operations in
F
q
m
, where
n
is the length of the code,
s
the interleaving order,
M
(
n
)
the complexity for multiplying two skew polynomials of degree at most
n
,
ω
the matrix multiplication exponent and
O
~
·
the soft-
O
notation which neglects log factors. This matches the previous best speeds for these tasks, which were obtained by top–down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial modules by the bottom-up KNH approach. In contrast to the top–down approach the bottom-up KNH algorithm has no requirements on the interpolation points and thus does not require any pre-processing. |
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| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0925-1022 1573-7586 |
| DOI: | 10.1007/s10623-023-01315-4 |