A Finite Field Fourier Transform for Vectors of Arbitrary Length
Finite field Fourier transforms are of great interest in coding and cryptography. They are, in particular, used for describing BCH and RS codes in the spectral domain and for representing the solutions of recurrence equations used in stream ciphers. So far, finite field Fourier transforms have only been defined on vectors that have a length which is relatively prime to the characteristic of the field. The aim of the paper is to generalize this definition to arbitrary lengths. Many properties get a simpler interpretation with this approach.
KeywordsDiscrete Fourier Transform Finite Field Cyclic Code Stream Cipher Binomial Coefficient
Unable to display preview. Download preview PDF.
- R.E. Blahut Theory and Practice of Error Control Codes Addison-Wesley Publishing Company, Inc., 1983.Google Scholar
- P. Mathys, “A generalization of the discrete Fourier transform in finite fields,” Proc. IEEE Symp. Inform. Theory San Diego (CA), Jan. 14–19, 1990.Google Scholar
- S.R. Blackburn, “A Generalization of the Discrete Fourier Transform: Determining the Minimal Polynomial of a Periodic Sequence,” IEEE Trans. Inform. Theory (to appear).Google Scholar
- C.G. Günther, “Fourier transform in cryptography and coding,” IEEE Workshop on Inform. Theory Bellagio, Italy, June 1987.Google Scholar
- H. Hasse, “Theorie der höheren Differentiale in einem algebraischen Funktionenkörper mit vollkommenem Konstantenkörper bei beliebiger Charakteristik,” J. reine angew. Math. Bd. 175, S. 50–54, 1936.Google Scholar
- L.M. Milne-Thomson The Calculus of Finite Differences MacMillan and Co., London, 1951.Google Scholar