Problems of Information Transmission, Vol. 39, No. 3, 2003, pp. 231–238. Translated from Problemy Peredachi Informatsii, No. 3, 2003, pp. 3–10.
Original Russian Text Copyright
2003 by Trifonov, Fedorenko.
INFORMATION THEORY AND CODING THEORY
A Method for Fast Computation of the
Fourier Transform over a Finite Field
P. V. Trifonov and S. V. Fedorenko
St. Petersburg State Polytechnic University
Received October 25, 2002; in ﬁnal form, February 5, 2003
Abstract—We consider the problem of fast computation of the Fourier transform over a ﬁnite
ﬁeld by decomposing an arbitrary polynomial into a sum of linearized polynomials. Examples of
algorithms for the Fourier transform with complexity less than that of the best known analogs
Presently, a number of fast Fourier transform (FFT) algorithms over the real or complex ﬁeld is
known, but translation of these algorithms to ﬁnite ﬁelds is not always possible. Furthermore, an
FFT algorithm specially constructed for a particular ﬁnite ﬁeld may be better than an algorithm
translated from another ﬁeld .
The suggested method consists in decomposing an original polynomial into a sum of linearized
polynomials (1) and evaluating them at a set of basis points (2). Components of the Fourier trans-
form are computed as linear combinations of these values with coeﬃcients from a prime ﬁeld (3).
An approach based on representing a polynomial as a sum of linearized ones was ﬁrst suggested
in  and then generalized in . In what follows, we consider basic notions and deﬁnitions,
introduce the cyclotomic decomposition of polynomials, and present an FFT algorithm based on
this decomposition. The algorithm is described for ﬁelds of characteristic 2 but can be generalized
for the case of an arbitrary ﬁnite ﬁeld.
2. BASIC NOTIONS AND DEFINITIONS
Deﬁnition 1. The Fourier transform of a polynomial f(x)=
of degree deg f (x)=n−1,
n | (2
− 1), in the ﬁeld GF (2
) is the collection of elements
= f (α
,j∈ [0,n− 1],
where α is an element of order n in the ﬁeld GF (2
Deﬁnition 2. A linearized polynomial over GF (2
) is a polynomial of the form
∈ GF (2
One can easily show that a linearized polynomial satisﬁes the equality L(a + b)=L(a)+L(b).
A consequence of this property is Theorem 1, which is presented here in a modiﬁed form.
2003 MAIK “Nauka/Interperiodica”