A new GCD algorithm for quadratic number rings with unique factorization

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(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.