The Discrete Fourier Transform in Coding and Cryptography

| Some applications of the Discrete Fourier Transform (DFT) in coding and in cryptography are described. The DFT over general commutative rings is introduced and the condition for its existence given. Blahut's Theorem, which relates the DFT to linear complexity, is shown to hold unchanged in general commutative rings. I. The (Usual) Discrete Fourier Transform Let be a primitive N th root of unity in a eld F , i.e., N = 1 but i 6= 1 for 1 i < N . The (usual) Discrete Fourier Transform (DFT) of length N generated by is the mapping DFT ( ) from F N to F de ned by B =DFT (b) in the manner