Carry Propagation in Multiplication by Constants

TL;DR

This paper analyzes carry propagation in constant multiplication, showing that the distribution of the longest carry chain in the final addition resembles that of adding two independent n-bit numbers, regardless of addition order.

Contribution

It demonstrates that the carry-propagation behavior in constant multiplication is statistically similar to standard addition, regardless of carry-save addition order.

Findings

Longest carry chain distribution matches that of adding two independent n-bit numbers.

Mean and variance of carry chain length are asymptotically the same as in standard addition.

Results hold for all carry-save addition sequences.

Abstract

Suppose that a random n-bit number V is multiplied by an odd constant M, greater than or equal to 3, by adding shifted versions of the number V corresponding to the 1s in the binary representation of the constant M. Suppose further that the additions are performed by carry-save adders until the number of summands is reduced to two, at which time the final addition is performed by a carry-propagate adder. We show that in this situation the distribution of the length of the longest carry-propagation chain in the final addition is the same (up to terms tending to 0 as n tends to infinity) as when two independent n-bit numbers are added, and in particular the mean and variance are the same (again up to terms tending to 0). This result applies to all possible orders of performing the carry-save additions.