Conversions between RNS and mixed-radix numbers using signed-digit arithmetic

By introducing signed-digit (SD) number arithmetic into a residue number system (RNS), arithmetic operations can be performed efficiently. In this paper, hardware algorithms converting the numbers in an RNS with a four-moduli set {2^P, 2^P-1 - 1, 2^P - 1, 2^P+1 - 1} to and from that in a mixed-radix number system (MRNS) by using SD arithmetic are proposed. The conversions between the RNS and MRNS numbers can be implemented using the binary tree structure of the fast modulo m SD adders, where m ϵ {2^P - 1, 2^P±1 - 1}. The modulo m SD addition is implemented simply uisng an end-around-carry SD adder, so that the addition time is independent of the word length of the operands and the conversion time is only dependent of the stages of additions. We also introduce the minimal SD number representation to reduce the stages of the binary addition tree structure. The comparison of the new converters using the SD number arithmetic with that using the binary arithmetic yields reductions in delay of 63% on average.