Lower bounds for decision problems in imaginary, norm-Euclidean quadratic integer rings

We prove lower bounds for the complexity of deciding several relations in imaginary, norm-Euclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, whic...

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Vydané v:Journal of symbolic computation Ročník 44; číslo 6; s. 683 - 699
Hlavný autor: Busch, J.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Elsevier Ltd 01.06.2009
Predmet:
Arithmetic complexity
Computational complexity
Coprimality
Eisenstein integers
Gaussian integers
gcd
Lower bounds
Coprimality
Lower bounds
Eisenstein integers
Gaussian integers
gcd
Arithmetic complexity
Computational complexity
ISSN:0747-7171, 1095-855X
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Popis
Shrnutí:We prove lower bounds for the complexity of deciding several relations in imaginary, norm-Euclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, which yield lower bounds for gcd computations. In each imaginary, norm-Euclidean quadratic integer ring, a known binary-like gcd algorithm has complexity that is quadratic in our lower bound.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2008.11.001