New transform using the Mersenne numbers

Number theoretic transforms (NTTs) find applications in the calculation of convolutions and correlations. They can perform these calculations without introducing additional noise in the processing due to rounding or truncation. Among all NTTs, Fermat and Mersenne number transforms have been given particular attention. However, the main drawback of these transforms is the inconvenient word length for the Fermat number transforms, and lack of fast algorithms for the Mersenne number transforms. The authors aim to introduce a new real transform defined modulo Mersenne numbers with long transform length equal to a power of two. This is achieved by dropping the condition that α_ should be ±2 and using a new definition for NTTs that departs from the usual Fourier-like definition. The new transform is suitable for fast algorithms. It has the cyclic convolution property and hence can be applied to the calculation of convolutions and correlations. The transform is extended to the two-dimensional case and then generalised to the multidimensional case. Examples are given for the one-dimensional and two-dimensional cases