Unusual-length number-theoretic transforms using recursive extensions of Rader's algorithm

Abstract

A novel decomposition of NTT blocklengths is proposed using repeated applications of Rader's algorithm to reduce the problem to that of realising a single small-length NTT. An efficient implementation of this small-length NTT is achieved by an initial basis conversion of the data, so that the new basis corresponds to the kernel of the small-length NTT. Multiplication by powers of the kernel become rotations and all arithmetic is efficiently performed within the new basis. More generally, this extension of Rader's algorithm is suitable for NTT or DFT applications where an efficient implementation of a particular small-length NTT/DFT module exists.