yeah, i just have a very visual brain. when you use these words i get pictures of processes, and i naturally see scales and vectors, symmetries and inversions. a lot of what seems absurd is just a reflection of what is sane. random number generators, for example. people like to split hairs about whether it's really random if you can model it, but taht's teh thing, you can't model it until after it's happened. it's because the main mechanism of PRNGs relates to rounding errors and overflows being sent around in a circle, asymmetric cryptography is entirely built on top of the entropy caused by clockwork arithmetic, which in practise just means that bits fall off the left and they are gone, but because of the permutations and the rules of arithmetic, you don't get zero straight away, you get less... something... which we call entropy. to make it secure, you need good entropy, and then it's hard to figure out what was lost because it was more complicated than the thing it was added to. haha. anyway. i love arithmetic. i first fell in love with it via the mandelbrot set but cryptography is even cooler. and the related coding systems, the principles of representation, integers, dimensions...

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Still have small issues with your framing of crypto, but can't get into it now. Another time! Oh, and one other thing I can't let go of, I think you once described finite fields inaccurately. They're entirely classsified, and all of order p^n for any n>=1