Binary GCD Like Algorithms for Some Complex Quadratic Rings

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Abstract

On the lines of the binary gcd algorithm for rational integers, algorithms for computing the gcd are presented for the ring of integers in \(\mathbb{Q}(\sqrt{d})\) where d ∈ { − 2, − 7, − 11, − 19}. Thus a binary gcd like algorithm is presented for a unique factorization domain which is not Euclidean (case d=-19). Together with the earlier known binary gcd like algorithms for the ring of integers in \(\mathbb{Q}(\sqrt{-1})\) and \(\mathbb{Q}(\sqrt{-3})\) , one now has binary gcd like algorithms for all complex quadratic Euclidean domains. The running time of our algorithms is O(n 2) in each ring. While there exists an O(n 2) algorithm for computing the gcd in quadratic number rings by Erich Kaltofen and Heinrich Rolletschek, it has large constants hidden under the big-oh notation and it is not practical for medium sized inputs. On the other hand our algorithms are quite fast and very simple to implement.