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A New GCD Algorithm for Quadratic Number Rings with Unique Factorization

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LATIN 2006: Theoretical Informatics (LATIN 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3887))

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Abstract

We present an algorithm to compute a greatest common divisor of two integers in a quadratic number ring that is a unique factorization domain. The algorithm uses \(O(n {\rm log}^{2} n {\rm log log} n + \Delta^ {\raisebox{0.8mm}{\scriptsize 1}{\scriptsize /}\raisebox{-0.5mm}{\scriptsize 2}} +^{\epsilon})\) bit operations in a ring of discriminant Δ. This appears to be the first gcd algorithm of complexity o(n 2) for any fixed non-Euclidean number ring. The main idea behind the algorithm is a well known relationship between quadratic forms and ideals in quadratic rings. We also give a simpler version of the algorithm that has complexity O(n 2) in a fixed ring. It uses a new binary algorithm for reducing quadratic forms that may be of independent interest.

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Agarwal, S., Frandsen, G.S. (2006). A New GCD Algorithm for Quadratic Number Rings with Unique Factorization. In: Correa, J.R., Hevia, A., Kiwi, M. (eds) LATIN 2006: Theoretical Informatics. LATIN 2006. Lecture Notes in Computer Science, vol 3887. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11682462_8

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  • DOI: https://doi.org/10.1007/11682462_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-32755-4

  • Online ISBN: 978-3-540-32756-1

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