ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 2, pp. 139–151.
Pleiades Publishing, Inc., 2006.
Original Russian Text
S.V. Fedorenko, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 2, pp. 81–93.
A Method for Computation of the Discrete
Fourier Transform over a Finite Field
S. V. Fedorenko
Saint-Petersburg State Polytechnic University
Received July 12, 2005; in ﬁnal form, March 9, 2006
Abstract—The discrete Fourier transform over a ﬁnite ﬁeld ﬁnds applications in algebraic
coding theory. The proposed computation method for the discrete Fourier transform is based
on factorizing the transform matrix into a product of a binary block circulant matrix and a
diagonal block circulant matrix.
We propose a method for computation of the discrete Fourier transform (DFT) over a ﬁnite
ﬁeld, based on factorizing the transform matrix into a product of a binary block circulant matrix
and a diagonal block circulant matrix. It is known that the asymptotically best algorithm for the
DFT computation is described by Wang and Zhu  and Afanasyev ; however, for small values
of the transform length, the cyclotomic algorithm  is most eﬃcient. The idea to reduce the
computation of the DFT of length n to the computation of a cyclic convolution of length n − 1
was proposed by Rader  for prime lengths. The Goertzel algorithm, modiﬁed for ﬁnite ﬁelds ,
is interpreted in  as a reduction of the DFT of length n =2
− 1 to the computation of cyclic
convolutions of length m. Bassalygo  was ﬁrst to draw attention to a regular structure of the
transform matrix, which makes it possible to apply block cyclic convolutions to the DFT compu-
tation. The algorithm of the present paper, having computation complexity not greater than that
of the cyclotomic algorithm , enjoys a simpler description.
2. BASIC NOTIONS AND DEFINITIONS
The discrete Fourier transform of length n of a vector f =(f
), i ∈ [0,n− 1], n | (2
− 1), in
the ﬁeld GF (2
) is the vector F =(F
,j∈ [0,n− 1],
where α is an element of order n in GF (2
). Let us write the DFT in a matrix form:
F = Wf,
where W =(α
), i, j ∈ [0,n− 1], is a Vandermonde matrix.
Consider the set of cyclotomic cosets modulo n over GF (2)
), ..., (c
(mod n), i ∈ [1,], being the number of cyclotomic cosets modulo n over GF (2).