Read Online Nine Algorithms That Changed the Future: The Ingenious Ideas That Drive Today's Computers by John MacCormick, Chris Bishop - Free Book Online Page B
Authors: John MacCormick, Chris Bishop
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daisy-yellow.
At this point, you and Arnold would like to give each other samples of your public-private mixtures, but remember it's against the rules to directly give a mixture of paint to one of the other people in the room. The only way to give a mixture to someone else is to make several batches of it and leave them in the middle of the room so that anyone who wants one can take it. This is exactly what you and Arnold do: each of you makes several batches of your public-private mixture and leaves them in the middle of the room. Eve can steal a batch or two if she wants, but as we will learn in a minute, they will do her no good at all. The figure on the following page shows the situation after this third step of the paint-mixing trick.
OK, now we're getting somewhere. If you think hard at this point, you might see the final trick that would allow you and Arnold to each create an identical shared secret mixture without letting Eve in on the secret. Here's the answer:
The paint-mixing trick, step 3: The public-private mixtures are available to anyone who wants them.
Step 4. You pick up a batch of Arnold's public-private mixture and take it back to your corner. Now add one pot of your private color. Meanwhile, Arnold picks up a batch of your public-private mixture and takes it back to his corner, where he adds it to a pot of his private color.
Amazingly, you have both just created identical mixtures! Let's check: you added your private color (lavender) to Arnold's public-private mixture (crimson and daisy-yellow), resulting in a final mixture of 1 lavender, 1 crimson, 1 daisy-yellow. What about Arnold's final mixture? Well, he added his private color (crimson) to your public-private mixture (lavender and daisy-yellow), resulting in a final mixture of 1 crimson, 1 lavender, 1 daisy-yellow. This is exactly the same as your final mixture. It really is a shared secret mixture. The figure on the next page shows the situation after this final step of the paint-mixing trick.
Now, what about Eve? Why can't she create a batch of this shared secret mixture? The reason is that she doesn't know your private color or Arnold's private color, and she needs at least one of them to create the shared secret mixture. You and Arnold have thwarted her, because you never left your private colors exposed, on their own, in the middle of the room. Instead, you each combined your private color with the public color before exposing it, and Eve has no way of “unmixing” the public-private mixtures to obtain a pure sample of one of the private colors.
The paint-mixing trick, step 4: Only you and Arnold can make the shared secret color, by combining the mixtures shown by arrows.
Thus, Eve has access only to the two public-private mixtures. If she mixes one batch of your public-private mixture with one batch of Arnold's public-private mixture, the result will contain 1 crimson, 1 lavender, and 2 daisy-yellow. In other words, compared to the shared secret mixture, Eve's mixture has an extra daisy-yellow. Her mixture is too yellow, and because there's no way to “unmix” paint, she can't remove that extra yellow. You might think Eve could get around this by adding more crimson and lavender, but remember she doesn't know your private colors, so she wouldn't know that these are the colors that need to be added. She can only add the combination of crimson plus daisy-yellow or lavender plus daisy-yellow, and these will always result in her mixture being too yellow.
Paint-Mixing with Numbers
If you understand the paint-mixing trick, you understand the essence of how computers establish shared secrets on the internet. But, of course, they don't really use paint. Computers use numbers, and to mix the numbers they use mathematics. The actual math they use isn't too complicated, but it's complicated enough to be confusing at first. So, for our next step toward understanding how shared secrets are established on the internet, we will use some
At this point, you and Arnold would like to give each other samples of your public-private mixtures, but remember it's against the rules to directly give a mixture of paint to one of the other people in the room. The only way to give a mixture to someone else is to make several batches of it and leave them in the middle of the room so that anyone who wants one can take it. This is exactly what you and Arnold do: each of you makes several batches of your public-private mixture and leaves them in the middle of the room. Eve can steal a batch or two if she wants, but as we will learn in a minute, they will do her no good at all. The figure on the following page shows the situation after this third step of the paint-mixing trick.
OK, now we're getting somewhere. If you think hard at this point, you might see the final trick that would allow you and Arnold to each create an identical shared secret mixture without letting Eve in on the secret. Here's the answer:
The paint-mixing trick, step 3: The public-private mixtures are available to anyone who wants them.
Step 4. You pick up a batch of Arnold's public-private mixture and take it back to your corner. Now add one pot of your private color. Meanwhile, Arnold picks up a batch of your public-private mixture and takes it back to his corner, where he adds it to a pot of his private color.
Amazingly, you have both just created identical mixtures! Let's check: you added your private color (lavender) to Arnold's public-private mixture (crimson and daisy-yellow), resulting in a final mixture of 1 lavender, 1 crimson, 1 daisy-yellow. What about Arnold's final mixture? Well, he added his private color (crimson) to your public-private mixture (lavender and daisy-yellow), resulting in a final mixture of 1 crimson, 1 lavender, 1 daisy-yellow. This is exactly the same as your final mixture. It really is a shared secret mixture. The figure on the next page shows the situation after this final step of the paint-mixing trick.
Now, what about Eve? Why can't she create a batch of this shared secret mixture? The reason is that she doesn't know your private color or Arnold's private color, and she needs at least one of them to create the shared secret mixture. You and Arnold have thwarted her, because you never left your private colors exposed, on their own, in the middle of the room. Instead, you each combined your private color with the public color before exposing it, and Eve has no way of “unmixing” the public-private mixtures to obtain a pure sample of one of the private colors.
The paint-mixing trick, step 4: Only you and Arnold can make the shared secret color, by combining the mixtures shown by arrows.
Thus, Eve has access only to the two public-private mixtures. If she mixes one batch of your public-private mixture with one batch of Arnold's public-private mixture, the result will contain 1 crimson, 1 lavender, and 2 daisy-yellow. In other words, compared to the shared secret mixture, Eve's mixture has an extra daisy-yellow. Her mixture is too yellow, and because there's no way to “unmix” paint, she can't remove that extra yellow. You might think Eve could get around this by adding more crimson and lavender, but remember she doesn't know your private colors, so she wouldn't know that these are the colors that need to be added. She can only add the combination of crimson plus daisy-yellow or lavender plus daisy-yellow, and these will always result in her mixture being too yellow.
Paint-Mixing with Numbers
If you understand the paint-mixing trick, you understand the essence of how computers establish shared secrets on the internet. But, of course, they don't really use paint. Computers use numbers, and to mix the numbers they use mathematics. The actual math they use isn't too complicated, but it's complicated enough to be confusing at first. So, for our next step toward understanding how shared secrets are established on the internet, we will use some
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