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, 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.