On the additive complexity of the cyclotomic FFT algorithm

The problem of efficient evaluation of the discrete Fourier transform over finite fields is considered. The techniques for additive complexity reduction of the cyclotomic FFT algorithm are proposed. The first one is based on the classical simultaneous reduction algorithm. The second one is based on a factorization of the presummation matrix into a sparse and block-diagonal ones. The proposed methods provide smaller asymptotic complexity, although for small-sized problems the required number of operations appears to be higher than the complexity of computer-optimized algorithms.