Hartley transforms over finite fields

A general f ramework is presented for constructing transforms in the field of the input which have a convolutionlike property. The construction is carried out over finite fields, but is shown to be valid over the real and complex fields as well. It is shown that these basefield transforms can be v iewed as “projections” of the discrete Fourier transform @IT) and that they exist for all lengths N for which the DFI is defined. The convolut ion property of the basefield transforms is der ived and a condit ion for such transforms to have the selfinverse property is given. Also, fast algorithms for these basefield transforms are developed, showing gains when compared to computat ions using the FIT. Application of the methodology to Hartley transforms over R leads to a simple derivation of fast algorithms for comput ing real Hartley transforms.