Fast Computation of the Biquadratic Residue Symbol

Abstract

This article describes an asymptotically fast algorithm for the computation of the biquadratic residue symbol. The algorithm achieves a running time of O( n(log n) 2log log n) for Gaussian integers bounded by 2 n in the norm. Our algorithm is related to an asymptotically fast GCD computation in Z [ i] which uses the technique of a controlled Euclidean descent in Z [ i]. At first, we calculate a Euclidean descent with suitable Euclidean steps x j−1 = q j x j + x j+1 storing the q j 's for later use. Then we calculate the biquadratic residue symbol of x 0, x 1 from the quotient sequence in linear time in the length of the q j 's.