Cryptography Part 1: Drunken rambling introduction

There is a series of posts forming in my head. I have no unifying theme nor particular audience in mind, so they will be even more rambling and incoherent than usual. Also I plan to have a drink or two before each just to complete the effect. You have been warned.

Let’s play a little game. You and I will be on the same team for a change. This is our asset:

It has 16-64 GB of storage, consumes 0.5 watts, and occupies 11.25 square inches.

On the bright side, we do have two of them. One each.

This is our adversary:

It has 3-12 exabytes of storage, consumes 65 megawatts, and occupies 100000 square feet. Also, it is operated by people smarter than you, smarter than me, and smarter than anybody either one of us has ever heard of.

The game is this. You and I will try to have a conversation over a great distance that is unintelligible to this adversary.

Now, perhaps it is just me — I still get excited by powered flight — but I find it awe inspiring that it might actually be possible for us to win. Not easily and not with certainty, but still.

That is what this series will be about. More or less.

Our story begins around 400 B.C. on the Greek island of Delos. Its citizens were suffering from internal strife threatening to tear the society apart. Or something. The island’s leaders consulted the Oracle at Delphi, who explained that Apollo was angry, and to appease him, the citizens had to construct a new altar double the size of their existing one.

Now, the ancient Greeks had far more rigorous minds than your typical modern engineer. To them, “construct” meant something very particular: Create something perfect using idealized versions of various masonry/carpentry tools. Extra credit for using only straightedge and compass.

(If you have never seen this game before, here are the rules. Given two points, you may use the straightedge to connect the points with a perfect line and extend it as far as you like. You may also set the compass to the distance between any two points, then draw a perfect circle with that radius centered on any other point. By starting with a few provided points and applying the straightedge and compass repeatedly, you generate new points at the intersections of all the lines and circles. That’s it. The ancient Greeks loved this stuff.)

Now, the altar to Apollo was a perfect cube. So the Delians started with a line segment AB having the same length as a side of the cube. Then they used a compass to draw a circle through A centered at B. Then they used the straightedge to extend AB to intersect the circle at C:

Since AC is twice AB, the Delians simply used that as the side of a new cubical altar.

But their problems only got worse. Eventually, they went back and asked the Oracle what was wrong. The Oracle explained that they had angered Apollo further by not following instructions, since they had created an altar not two times but eight times the original’s volume. Apparently, gods can be picky.

The Greeks eventually solved this problem by adding various interesting contraptions to their idealized toolkit. But the extra credit problem remained: Given a segment of length 1, can you construct one of length \(\sqrt[3]{2}\) using only straightedge and compass?

This problem stumped geometers and would-be geometers for several years. Two thousand, actually. That is how long humanity needed to develop the mathematical tools to solve this Delian problem, as it came to be known. What that solution was and how it relates to anything will be the topic for…

Next time: Gauss, Galois, et. al.

(I did try to warn you.)

3 comments to Cryptography Part 1: Drunken rambling introduction

  • Nice intro.

    Not sure the Greeks were any more idealised in their engineering; I think that’s just our maths teachers handing it down. Just pretend they needed to figure out how much stuff to order in order to make budget projections or for some other reason they wanted to get the answer right before trying a bunch of stuff.

    Recently read this somewhere and it seems apropos:
    “Use an insecure method of communication once. Die once.”

  • I’m also not sure you can measure BC or AB in the geometric construction. As I understand it the circles and straight lines simply mean you can get [a] a fixed distance, equal to some other length you’ve already made, or [b] a fixed angle. But if they could just divide by two why wouldn’t they divide by √3 or √(2+√5) ?

  • from Wikipedia:

    According to Plutarch, Plato gave the problem to Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry (Plut., Quaestiones convivales VIII.ii, 718ef).

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