Linear cryptanalysis is widely known as one of the fundamental tools for the crypanalysis of block ciphers. Over the decades following its first introduction by Matsui in [Ma94a], many different extensions and improvements have been proposed. One of them is [CSQ07], where Collard et al. use the Fast Fourier Transform (FFT) to accelerate the parity computations which are required to perform a linear key recovery attack. Modified versions of this technique have been introduced in order to adapt it to the requirements of several dedicated linear attacks. This work provides a model which extends and improves these different contributions and allows for a general expression of the time and memory complexities that are achieved. The potential of this general approach will then be illustrated with new linear attacks on reduced-round PRESENT, which is a very popular and widely studied lightweight cryptography standard. In particular, we show an attack on 26 or 27-round PRESENT-80 which has better time and data complexity than any previously known attacks, as well as the first attack on 28-round PRESENT-128.