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 DFT into several circular convolutions, and then derives the DFT results from the circular convolution results. When applied to a DFT over GF(2m), 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.
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 GF(pm) is defined as
where is the N-th primitive root of 1 in GF(pm). If we define the polynomial representation of 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 DFT has an complexity. Fast Fourier transforms are just efficient algorithms evaluating the above matrix-vector product.
First, we define a linearized polynomial over GF(pm) as
is called linearized because , which comes from the fact that for elements
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, and hence is given by
Expanding with the proper basis , we have where , and by the property of the linearized polynomial , we have
This equation can be rewritten in matrix form as , where is an matrix over GF(p) 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 we successfully reduce the discrete Fourier transform into short convolutions.
When applied to a characteristic-2 field GF(2m), the matrix is just a binary matrix. Only addition is used when calculating the matrix-vector product of and . It has been shown that the multiplicative complexity of the cyclotomic algorithm is given by , and the additive complexity is given by .
- S.V. Fedorenko and P.V. Trifonov, Fedorenko, S. V.; Trifonov, P. V.. (2003). "On Computing the Fast Fourier Transform over Finite Fields". Proceedings of International Workshop on Algebraic and Combinatorial Coding Theory: 108–111.
- Wu, Xuebin; Wang, Ying; Yan, Zhiyuan (2012). "On Algorithms and Complexities of Cyclotomic Fast Fourier Transforms Over Arbitrary Finite Fields". IEEE Transactions on Signal Processing 60 (3): 1149–1158. doi:10.1109/tsp.2011.2178844.