Finite Fields

So when you ask "when will finite field arithmetic be useful?", that’s when. In the 70s, there were gangs of raiders. Paintball, mainly, but that still stings.

So they’d wander from door to door, pretty much dropping in at random. Drive-by paintball, I guess you could call it, but they didn’t always drive.

The thing is, they would let you off if you could do the maths. Not always finite field arithmetic, but that was definitely the biggest area. "Let ω be a cube root of 5 adjoined to a prime field of characteristic 7,…" and all that. Then it was all calculating inverses of arbitrary elements.

Not always cubes, either! They were allowed to go up to degree 9 or 10, I think - the law was different back then. But the primes weren’t normally massive. I know that they did 11, and I heard tell of a 13 once, but I’m not convinced.

So you’re sat at home, and all of a sudden you’ve got marauders asking you to calculate the inverse of 3+5ω inside some cubic extension of Z7, with a paintball gun pressed up against your nose. Tough times.

Tom put his hand up, and I nodded. "But that’s all pencil and paper stuff!" It wasn’t a question, but you could hear it from the tone.

You could have a pencil and paper, but the pencil was normally blunt. This stuff just wasn’t policed at all! You could make scratch marks, but that’s not enough! You weren’t there. You can’t understand how hard it is to do this kind of thing without a decent pencil - it’s more of a hindrance than a help! You end up just looking at squiggles on the paper and trying to remember if you’re written a European 7 or a minus sign!

You have to be able to do the mental arithmetic! Otherwise you’ve got 30 seconds of omegas and eights, and then a paintball bruise the size of a duck egg on your cheekbone!

Tom chimed in again. "But that’s what running training is for!"

Oh, Tom! They’re in your house at this point. Where could you run? Upstairs? Into the kitchen? You’re not in a forest with trees to hide behind! They’ve got backup too - a guy at the door with a master’s degree in computational complexity. Then what? Is that what you want?

He looked stricken. "We haven’t covered computational complexity yet! Isn’t it all just log(n)?"

I had a lot to cover, it seemed. I tried to calm things down a bit.

I know, Tom, I know. Try not to worry. Anyway, this isn’t the 70s - the rules are different now. They can’t just come into your house with a paintball gun. It’s all warrants and due diligence.

"My dad says they used to do the periodic table too."

They did, Tom, they did. But that’s not my area. And the memory stuff is much easier. And if you’re safe at home, indoors, who’s to say you can’t just have a massive copy of it on the wall? They can’t stop you - just say you were using it for revision. They love that.


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