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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Bibliographic Details
Published in:Combinatorics, probability & computing Vol. 31; no. 1; pp. 166 - 183
Main Authors: Giménez, Nardo, Matera, Guillermo, Pérez, Mariana, Privitelli, Melina
Format: Journal Article
Language:English
Published: Cambridge Cambridge University Press 01.01.2022
Subjects:
Algorithms
Asymptotic properties
Combinatorics
Complexity
Fields (mathematics)
Polynomials
ISSN:0963-5483, 1469-2163
Online Access:Get full text
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Summary: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.
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ISSN:0963-5483
1469-2163
DOI:10.1017/S0963548321000274