Application of symmetric redundant residues for fast and reliable arithmetic

Despite difficulties in general division, magnitude comparison, and sign detection, residue number system arithmetic has been used for many special-purpose systems in light of its parallelism and modularity for the most common arithmetic operations of addition/subtraction and multiplication. Computation in RNS requires modular reduction, both for the initial conversion from binary to RNS and after each operation to bring the result back to within a valid residue range. Use of redundant residues simplifies this critical operation, leading to even faster arithmetic. One type of redundant mod-m residue, that keeps the representational redundancy to the minimum of 1 bit per residue, has the nearly symmetric range [–m, m) and allows two values for each pseudoresidue: 〈x〉m or 〈x〉m – m. We study the extent of simplification and speed-up in the modular reduction process afforded by such redundant residues and discuss its potential implications to the design of RNS arithmetic circuits. In particular, we show that besides cost and performance benefits, introduction of error checking and fault tolerance in arithmetic computations is facilitated when such redundant residues are used.