Cyclotomic fast Fourier transform
The cyclotomic fast Fourier transform is a type of fast Fourier transform algorithm over finite fields. This algorithm first decomposes a discrete Fourier transform into several circular convolutions. This then derives the discrete Fourier transform results from the circular convolution results. When applied to a discrete Fourier transform over, this algorithm has a very low multiplicative complexity. In practice, since there usually exist efficient algorithms for circular convolutions with specific lengths, this algorithm is very efficient.
Background
The discrete Fourier transform over finite fields finds widespread application in the decoding of error-correcting codes such as BCH codes and Reed–Solomon codes. Generalized from the complex field, a discrete Fourier transform of a sequence over a finite field is defined aswhere is the N-th primitive root of 1 in. If the polynomial representation of can defined as
it is easy to see that is simply. That is, the discrete Fourier transform of a sequence converts it to a polynomial evaluation problem.
Written in matrix format,
Direct evaluation of the discrete Fourier transform has an complexity. Fast Fourier transforms are just efficient algorithms evaluating the above matrix-vector product.
Algorithm
First, define a linearized polynomial over asHere is called linearized, because, which comes from the fact that for elements and.
Notice that is invertible modulo because must divide the order of the multiplicative group of the field. So, the elements can be partitioned into cyclotomic cosets modulo :
where. Therefore, the input to the Fourier transform can be rewritten as
In this way, the polynomial representation is decomposed into a sum of linear polynomials. Hence, is given by
.
Expanding with the proper basis yields, where. By the property of the linearized polynomial, this yields
This equation can be rewritten in matrix form as, where is an matrix over that contains the elements, is a block diagonal matrix, and is a permutation matrix regrouping the elements in according to the cyclotomic coset index.
Note that if the normal basis is used to expand the field elements of, the i-th block of is given by:
which is a circulant matrix. It is well known that a circulant matrix-vector product can be efficiently computed by convolutions. Hence, it is successful to reduce the discrete Fourier transform into short convolutions.