The so-called affine projection algorithm (APA) has become a popular method in adaptive filtering applications and fast versions of it have been developed previously, such as fast affine projection (FAP) and the frequency domain block exact fast affine projection (BEFAP). While it is known that regularization is an absolute necessity in affine projection based adaptive filtering algorithms, FAP and BEFAP rely on an implicit "small regularization parameter" assumption. In this paper, a block exact affine projection algorithm (BE-APA) is derived that does not rely on the assumption of a small regularization parameter. It is an exact frequency domain translation of the original AP algorithm, and still has about the same complexity as, block exact FAR Unlike BEFAP, it can be used with strong regularization. It is then extended to incorporate an alternative to explicit regularization that is based on so-called "sparse" equations.