Carry-save adder

A carry-save adder is a type of digital adder, used in computer microarchitecture to compute the sum of three or more n-bit numbers in binary. It differs from other digital adders in that it outputs two numbers of the same dimensions as the inputs, one which is a sequence of partial sum bits and another which is a sequence of carry bits.

Consider the sum:

Using basic arithmetic, we calculate right to left, "8+2=0, carry 1", "7+2+1=0, carry 1", "6+3+1=0, carry 1", and so on to the end of the sum. Although we know the last digit of the result at once, we cannot know the first digit until we have gone through every digit in the calculation, passing the carry from each digit to the one on its left. Thus adding two n-digit numbers has to take a time proportional to n, even if the machinery we are using would otherwise be capable of performing many calculations simultaneously.

In electronic terms, using bits (binary digits), this means that even if we have n one-bit adders at our disposal, we still have to allow a time proportional to n to allow a possible carry to propagate from one end of the number to the other. Until we have done this,

A carry look-ahead adder can reduce the delay. In principle the delay can be reduced so that it is proportional to logn, but for large numbers this is no longer the case, because even when carry look-ahead is implemented, the distances that signals have to travel on the chip increase in proportion to n, and propagation delays increase at the same rate. Once we get to the 512-bit to 2048-bit number sizes that are required in public-key cryptography, carry look-ahead is not of much help.

The idea of delaying carry resolution until the end, or saving carries, is due to John von Neumann.

Here is an example of a binary sum:

Carry-save arithmetic works by abandoning the binary notation while still working to base 2. It computes the sum digit by digit, as

The notation is unconventional but the result is still unambiguous. Moreover, given n adders (here, n=32 full adders), the result can be calculated after propagating the inputs through a single adder, since each digit result does not depend on any of the others.

If the adder is required to add two numbers and produce a result, carry-save addition is useless, since the result still has to be converted back into binary and this still means that carries have to propagate from right to left. But in large-integer arithmetic, addition is a very rare operation, and adders are mostly used to accumulate partial sums in a multiplication.

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