Free Nine Algorithms That Changed the Future: The Ingenious Ideas That Drive Today's Computers by John MacCormick, Chris Bishop Page A

about it the right way, this isn't amazing at all. When Arnold and you managed to both produce the same shared secret color, it was because you mixed together the same three original colors, but in a different order: each of you kept one of the colors private, combining it with a publicly available mixture of the other two. The same thing has happened here with numbers. You both arrived at the same shared secret by multiplying together the same three numbers: 4, 6, and 7. (Yes, as you can check for yourself, 4 × 6 × 7 = 168.) But you arrived at the shared secret by keeping 4 private and “mixing” (i.e., multiplying) it with the publicly available mixture of 6 and 7 (i.e., 42) that had been announced by Arnold. On the other hand, Arnold arrived at the shared secret by keeping 6 private and mixing it with the publicly available mixture of 4 and 7 (i.e., 28) that you had announced.
    Just as we did in the paint-mixing trick, let's now verify that Eve has no chance of working out the shared secret. Eve gets to hear the value of each public-private number as it is announced. So she hears you say “28,” and Arnold say “42.” And she also knows the public number, which is 7. So if Eve knew how to do division, she could work out all your secrets immediately, just by observing that 28 ÷ 7 = 4, and 42 ÷ 7 = 6. And she could go on to compute the shared secret by calculating 4 × 6 × 7 = 168. However, luckily, we are using pretend math in this game: we assumed that multiplication was a one-way action and therefore Eve doesn't know how to divide. So she is stuck with the numbers 28, 42, and 7. She can multiply some of them together, but that doesn't tell her anything about the shared secret. For example, if she takes 28 × 42 = 1176, she is way off. Just as in the paint-mixing game her result was too yellow, here her result has too many 7's. The shared secret has only one factor of 7 in it, since 168 = 4 × 6 × 7. But Eve's attempt at cracking the secret has two factors of 7 in it, since 1176 = 4 × 6 × 7 × 7. And there's no way she can get rid of that extra 7, since she doesn't know how to do division.

    The number-mixing trick, step 4: Only you and Arnold can make the shared secret number, by multiplying together the numbers shown by arrows.
    Paint-Mixing in Real Life
    We have now covered all of the fundamental concepts needed for computers to establish shared secrets on the internet. The only flaw in the paint-mixing-with-numbers scheme is that it uses “pretend math,” in which we pretended that none of the parties could do division. To complete the recipe, we need a real-life math operation that is easy to do (like mixing paint) but practically impossible to undo
    (like unmixing paint). When computers do this in real life, the mixing operation is a thing called discrete exponentiation and the unmixing operation is called the discrete logarithm. And because there is no known method for a computer to calculate discrete logarithms efficiently, discrete exponentiation turns out to be just the kind of oneway action we are looking for. To explain discrete exponentiation properly, we're going to need two simple mathematical ideas. And we'll also need to write a few formulas. If you don't like formulas, just skip the rest of this section—you already understand almost everything about this topic. On the other hand, if you really want to know how computers do this magic, read on.
    The first important math idea we need is called clock arithmetic. This is actually something we are all familiar with: there are only 12 numbers on a clock, so every time the hour hand goes past 12, it starts counting again from 1. An activity that starts at 10 o'clock and lasts 4 hours finishes at 2 o'clock, so we might say that 10 + 4 = 2 in this 12-hour clock system. In mathematics, clock arithmetic works the same way, except for two details: (i) the size of the clock can be any number (rather than the familiar 12 numbers on a regular clock),