VLSI Costs of Arithmetic Parallelism: A Residue Reverse Conversion Perspective

This paper reports how VLSI cost metrics (area, delay, power) of residue reverse converters scale with the cardinality and dynamic range of moduli sets. The study uses CMAC reverse converters, reported previously by the authors to be the most efficient known to date in terms of area and delay. In all, 134 reverse converters with dynamic ranges from 32 to 120 bits and set cardinalities ranging from 4 to 20 are actually constructed and analyzed. It is seen that area, delay and power costs are cardinality insensitive once the cardinality exceeds a threshold (usually between five to eight). For cardinalities beyond this threshold, conversion costs are essentially dynamic range dependent. This insensitivity is explained in detail by noting the counterbalancing effects of the various sub-units of a CMAC reverse converter. Since practical implementations of RNS usually employ cardinalities beyond the abovementioned thresholds, the significance of this study is its conclusion ! that increasing the set cardinality in most implementations will have a marginal, if any, effect on VLSI reverse conversion costs.