We propose SWIFFT, a collection of compression functions that are highly parallelizable and admit very efficient implementations on modern microprocessors. The main technique underlying our functions is a novel use of the Fast Fourier Transform (FFT) to achieve "diffusion," together with a linear combination to achieve compression and "confusion." We provide a detailed security analysis of concrete instantiations, and give a high-performance software implementation that exploits the inherent parallelism of the FFT algorithm. The throughput of our implementation is competitive with that of SHA-256, with additional parallelism yet to be exploited. Our functions are set apart from prior proposals (having comparable efficiency) by a supporting asymptotic security proof: it can be formally proved that finding a collision in a randomly-chosen function from the family (with noticeable probability) is at least as hard as finding short vectors in cyclic/ideal lattices in the worst case.