This paper is about a new efficient method for the implementation of a Block Proportionate Normalized Least Mean Square (BPNLMS++) adaptive filter using the Fermat Number Transform (FNT) and its inverse (IFNT). These transforms present advantages compared to Fast Fourier Transform (FFT) and the inverse (IFFT). An efficient state space method for implementing the FNT over rectangular windows is used in the cases where there is a large overlap between the consecutive input signals. This is called Generalized Sliding Fermat Number Transform (GSFNT) and is useful for reducing the computational complexity of finite ring convolvers and correlators. In this contribution, we propose, as a first objective, an efficient state algorithm with the purpose of reducing the complexity of IFNT. This algorithm, called Inverse Generalized Sliding Fermat Number Transform (IGSFNT), uses the technique of Generalized Sliding associated to matricial calculation in the Galois Field. The second objective is to realize an implementation of the BPNLMS++ adaptive filter using GSFNT and IGSFNT, which can significantly reduce the computation complexity of the filter implantation on digital signal processors.