Speedup of discrete Fourier transform by efficient modular arithmetic

Abstract

The fast Fourier transform (FFT) based on modular arithmetic can compute convolution without round-off errors, which is desirable in many applications such as computational algebra and combinatory pattern matching. One of the critical challenges of the FFT is to enhance the performance. An effective approach is to optimize the high-cost operations. Modular reduction is one of the most frequently used high-cost operations that is a bottleneck of the FFT using modular arithmetic. In this article, we present three modular reduction methods and apply them to the implementation of the FFT.We use the strategy of delaying the modular reduction in the first method. We apply the Montgomery reduction to the FFT in the second method. The two methods both first transform the input, and then replace the modular reductions with lightweight replacements, and apply the reverse transform in the output stage to compute the right results.In the third method, we design an efficient modular reduction for the specific form of modular used in FFT. Experiments show that the incorporation of the new modular reductions speedups the FFT based on modular arithmetic.