Asymptotically fast methods for exact computations have been known for a quarter of a century. Unfortunately their impact on computer algebra systems has been reduced since it was believed that they were irrelevant in practice.
This myth died in the recent years and these methods permitted to overcome the size of magnitude of effective computations in several areas like exact factorization of polynomials.
As we shall see in this series of lectures, the successful implementation of these methods involve elegant algorithms, careful programing and accurate experimentation. Among the presented topics will be the methods based on the fast Fourier transform, the Newton iteration, the fast Chinese remaindering and fast linear algebra.