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 log2 n log log n + Δ1/2+ε) bit operations in a ring of discriminant Δ. This appears to be the first gcd algorithm of complexity o(n2) 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(n2) in a fixed ring. It uses a new binary algorithm for reducing quadratic forms that may be of independent interest. © Springer-Verlag Berlin Heidelberg 2006.
CITATION STYLE
Agarwal, S., & Frandsen, G. S. (2006). A new GCD algorithm for quadratic number rings with unique factorization. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 3887 LNCS, pp. 30–42). Springer Verlag. https://doi.org/10.1007/11682462_8
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