私は最近質問を受け、"how do you multiply without using the multiplication operator, without any sort of looping statements or explicit addition"ビット単位の操作にまったく慣れていないことに気付きました。
明らかにウィキペディアがありますが、初心者向けの説明がもっと必要です。このハックガイドもありますが、まだ把握できるレベルではありません。
私は Safari Books やその他のリソースを通じて優れたライブラリにアクセスできるので、本の章を指摘していただいてもかまいません。
Knuth、第 2 巻 - 半数値アルゴリズム
The crux of this comes down to a "half adder" and a "full adder". A half adder adds two bits of input to produce a single-bit result, and a single-bit carry. A full adder adds three bits of input (two normal inputs plus a carry from a lower bit) to produce a single-bit result and a single-bit carry.
In any case, the result is based on a truth table for addition. For a half adder, that is: 0+0=0, 0+1=1, 1+0=1, 1+1=0+carry.
So, the "normal" part of the result is the XOR of the inputs. The "carry' part of the result is the AND of the inputs. A full adder is pretty much the same, but left as the infamous "exercise for the reader".
Putting those together, you use a half-adder for the least significant bit, and full adders for the other bits to add N bits of input.
Once you can do addition, there are a couple of ways of doing multiplication. The easy (and slow) way to multiply NxM is to add N to itself M times. The faster (but somewhat more difficult to understand) way is to shift and add. For example, Nx5 = Nx4 + Nx1. You can produce NxB, where B = 2L by shifting N left by L bits.