On Modulus Replication for Residue Arithmetic Computations of Complex Inner Products

Residue Number Systems require the selection of ring moduli whose product is greater than the predicted dynamic range of the computation being performed. The restriction that the moduli be relatively prime usually limits the set of available moduli and hence the maximum dynamic range. This is particularly the case when small moduli are to be considered for efficient hardware implementation. Severe restrictions occur when algebraic constraints, such as those posed by the necessity to implement quadratic residue rings, are a factor. This paper presents a technique for coding weighted magnitude components (e.g. bits) of numbers directly into polynomial residue rings, such that repeated use may be made of the same set of moduli to effectively increase the dynamic range of the computation. This effectively limits the requirement for large sets of relatively prime moduli. For practical computations over quadratic residue rings, at least 6-bit moduli have to be considered; we show, in this paper, that 5-bit moduli can be effectively used for large dynamic range computations.