Convolutions over residue classes of quadratic integers

A Fourier-like transform is defined over a ring of quadratic integers modulo a prime number q in the quadratic field R(\sqrt{m}) , where m is a square-free integer. If q is a Fermat prime, one can utilize the fast Fourier transform (FFT) algorithm over the resulting finite fields to yield fast convolutions of quadratic integer sequences in R(\sqrt{m}) . The theory is also extended to a direct sum of such finite fields. From these results, it is shown that Fourier-like transforms can also be defined over the quadratic integers in R( \sqrt{m}) modulo a nonprime Fermat number.