A New Algorithm for Carry-Free Addition of Binary Signed-Digit Numbers

Signed-digit (SD) numbers generalize traditional radix numbers by allowing negative digits within a certain range. Typically, this leads to redundant number representations that can be used to avoid the carry propagation problem of addition of radix numbers. Unfortunately, as proved by Avizienis, the standard algorithm for carry-free addition of SD numbers does not work for the binary case. In this paper, we therefore construct a special algorithm for the carry-free addition and subtraction of binary SD numbers, i.e., addition and subtraction of n-digit numbers are performed with circuits of depth O(1) and size O(n). This is possible by computing in addition to the transfer digits used by the standard algorithm one additional bit that allows us to distinguish relevant cases to avoid propagation of dependencies. The additional bit and the transfer digit used to compute the sum digit at position i depend only on the summands' digits at positions i and i - 1 so that all sum digits can be computed with a hardware circuit of a depth that is independent of the number of digits. We first explain the basics of the standard addition algorithm to derive the additional information needed to fix the algorithm for the binary case. After proving the correctness of our algorithm, we present experimental results that show that our implementation clearly outperforms two's complement addition even for small numbers, and saves 50% of the required chip area compared to other carry-free implementations.