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The authors investigate the relationship between the Walsh-Hadamard transform (WHT) and the Fermat number transform (FNT). An interesting connection between the two transforms is revealed. This had led to the derivation of a block diagonal matrix which allows the computation of the FNT from the WHT and vice versa. The number of operations required to calculate the two transforms is less than is required when the direct method is used to calculate only the FNT. Moreover, this matrix can be implemented using only shifts and additions or only additions and small lookup tables for any kernel; it is also suitable for VLSI implementation and parallel processing.