Fast algebraic convolution for prime power lengths

In this paper a new fast convolution algorithm is introduced. The concept of this new algorithm is fully algebraic. Therefore, no roundoff errors occur. The signal length N is a prime power N = pⁿ, p ≥ 2. Hence the paper generalizes the results of a previous paper [17] for power of 2 lengths. The cyclic convolution of signals of length N is interpreted as a multiplication of polynomials of degree N - 1 modulo the polynomial X^N - 1. The calculation in our new method is done in a finite field extension of the field Q of rational numbers. To minimize the number of operations, an extension field Q[ω] is chosen, with ω as a symbolic root of unity and deg[Q[ω]:Q]=(p-1/p ⌈√N⌉^·). Here. ⌈⌉^· and ⌊⌋^· denote the well-known CEILING and FLOOR functions, respectively, but rather a mapping to the next larger power of p. The method achieves an arithmetic cost of O(N log N-log log TV) operations over Q. which is the same as that of the well- known algorithm by Schönhage-Strassen. However, our algorithm avoids the much more complicated FERMAT arithmetic over integers and works for primes p ≠ 2 as well, although it performs best with small p, preferably p = 2 or 3.