Average-case complexity of the Euclidean algorithm with a fixed polynomial over a finite field
We analyse the behaviour of the Euclidean algorithm applied to pairs ( g , f ) of univariate nonconstant polynomials over a finite field $\mathbb{F}_{q}$ of q elements when the highest degree polynomial g is fixed. Considering all the elements f of fixed degree, we establish asymptotically optimal b...
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| Vydáno v: | Combinatorics, probability & computing Ročník 31; číslo 1; s. 166 - 183 |
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| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cambridge
Cambridge University Press
01.01.2022
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| Témata: | |
| ISSN: | 0963-5483, 1469-2163 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We analyse the behaviour of the Euclidean algorithm applied to pairs (
g
,
f
) of univariate nonconstant polynomials over a finite field
$\mathbb{F}_{q}$
of
q
elements when the highest degree polynomial
g
is fixed. Considering all the elements
f
of fixed degree, we establish asymptotically optimal bounds in terms of
q
for the number of elements
f
that are relatively prime with
g
and for the average degree of
$\gcd(g,f)$
. We also exhibit asymptotically optimal bounds for the average-case complexity of the Euclidean algorithm applied to pairs (
g
,
f
) as above. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0963-5483 1469-2163 |
| DOI: | 10.1017/S0963548321000274 |