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Average-case complexity of the Euclidean algorithm with a fixed polynomial over a finite field

Part of: Algorithms - Computer Science General field theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  06 July 2021

Nardo Giménez ,
Guillermo Matera ,
Mariana Pérez  and
Melina Privitelli
Nardo Giménez
Affiliation:
Universidad Nacional de General Sarmiento, Instituto del Desarrollo Humano, J.M. Gutiérrez 1150 (B1613GSX) Los Polvorines, Buenos Aires, Argentina
Guillermo Matera*
Affiliation:
Universidad Nacional de General Sarmiento, Instituto del Desarrollo Humano, J.M. Gutiérrez 1150 (B1613GSX) Los Polvorines, Buenos Aires, Argentina Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Matemática, Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina
Mariana Pérez
Affiliation:
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina Universidad Nacional de Hurlingham, Instituto de Tecnología e Ingeniería, Av. Gdor. Vergara 2222 (B1688GEZ), Villa Tesei, Buenos Aires, Argentina
Melina Privitelli
Affiliation:
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina Universidad Nacional de General Sarmiento, Instituto de Ciencias, J.M. Gutiérrez 1150 (B1613GSX) Los Polvorines, Buenos Aires, Argentina
*
*Corresponding author. Emails:gmatera@campus.ungs.edu.ar, materaguillermo@gmail.com
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Abstract

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.

Keywords

Finite fieldsrational pointsEuclidean algorithmsymmetric functionsresultantaverage-case complexity

MSC classification

Primary: 12E20: Finite fields (field-theoretic aspects)
Secondary: 14G05: Rational points 14G15: Finite ground fields 68W40: Analysis of algorithms
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 1 , January 2022 , pp. 166 - 183
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

The authors were partially supported by the grants PIP CONICET 11220130100598 and PIO CONICET-UNGS 14420140100027

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