An Extended Euclid Algorithm for Rational Reconstruction

Let n/d∈Q,m be a positive integer and Let u=n/d mod m. Thus u is the image of a rational number modulo m. The rational reconstruction problem is: given u and m find n/d. Classical Euclidean Algorithm outputs n/d when m>;2M2, where M=max(|n|,d). The rational reconstruction problem was generally solved by classical Euclidean algorithm. In this paper, we achieve an Extended Euclidean algorithm for rational, and obtain an solvability criterion, hence, the number of solutions. Its Computational complexity is O(mlog2mloglogm).