In this thesis, a detailed analysis of sufficient conditions for existence of unique multidimensional linear and multidimensional non-linear Index map has been presented, along with a new Index representation. The recent Ideas of converting Discrete Fourier Transform to convolution and Implementing convolution efficiently, have been combined to give two algorithms viz. Nested Fourier Algorithm (NFA -- using linear multidimensional map) and Index Fourier Algorithm (IFA using a non-linear Index map). The two algorithms have been compared for the amount of arithmetic computations required. The algorithms have been Implemented In FORTRAN on IBM 37/155 and their execution timings have been compared.