On Algorithms and Complexities of Cyclotomic Fast Fourier Transforms Over Arbitrary Finite Fields

Discrete Fourier transforms over finite fields are significant due to their widespread applications in cryptography and error control codes, which in turn are used in all digital communication and storage systems. Cyclotomic fast Fourier transforms (CFFTs) are of great interest due to their low multiplicative complexities. However, all existing CFFTs are for characteristic-2 fields, and the computational complexities of CFFTs have not been analyzed theoretically. This paper addresses both problems for CFFTs, and has three main contributions to this end. First, we propose an efficient bilinear algorithm to compute Toeplitz matrix vector products (TMVPs), which has a lower computational complexity than existing algorithms, and works on all finite fields as well as the real and complex fields. Second, we propose an efficient algorithm for cyclic convolutions over arbitrary finite fields, which is essential in deriving efficient CFFTs over arbitrary finite fields. Finally, we derive bounds on the additive and multiplicative complexities of CFFTs over arbitrary finite fields. Our results confirm that CFFTs have the smallest multiplicative complexities among all known algorithms. Although their high additive complexities render them asymptotically suboptimal, CFFTs remain valuable since they have the smallest overall complexities for DFTs of up to thousands of symbols in most cases.