CS341 Study Guide - Andrey Kolmogorov, Machine Code, Memory Stick
Document Summary
One of the most fundamental problems you can imagine is multiplication. Computers all over the world are multiplying billions of numbers per second to do things like weather prediction, web ranking, and so forth. For small numbers that can fit into the word size, you use the built-in machine instruction. Cryptosystems such as rsa crucially depend on this kind of thing. The obvious brute-force pencil and paper method uses o(n2) time on two n-bit inputs. Let us look at karatsuba"s 1960-2 algorithm for multiplying two n bit numbers in o(n1. 59) steps. (here, by the way, we are using the log-cost model, but pretending that there is no multiplication instruction in our ram. ) Karatsuba"s algorithm is a great algorithm because (i) it is simple; (ii) it is actually useful in practice for large numbers and (iii) it really beats the naive pencil-and-paper algorithm, even on relatively small numbers.
