We propose two families of scalable hash functions for collision-resistant hashing that are highly parallel and based on the generalized fast Fourier transform (FFT). FFT-hashing is based on multipermutations. This is a basic cryptographic primitive for perfect generation of diffusion and confusion which generalizes the boxes of the classic FFT. The slower FFT-hash functions iterate a compression function. For the faster FFT-hash functions all rounds are alike with the same number of message words entering each round.