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It'll need some category theory, but I'll try.

So the question was, why elliptic curves are in some sense the "most abstract" group possible for Diffie-Hellman style dlog based schemes. For this we need to look at the dlog, and its inverse, the canonical map of ℤ into any given group with a marked element (the generator)

That canonical map maps an integer to the generator applied to itself the given number of times (for negative integers, we take the inverse). Canonical means that given the generator, and the information that 1 is mapped to it the map cannot have any other form.

The more closely I read the paper, the worse it gets. I'm gonna stop now.

To elaborate: after having read this in two version paper, a German master thesis that seems to be talking about the same algorithm without any of the outrageous claims, but the same performance numbers for factoring 10^20, I mostly think it's some sort of mistake, but 🤷🏼♀️

There is some other weirdness to the various papers, like explaining triple L in an amount of detail that is somewhat unusual for a research paper, while completely glossing over other parts of the claims.

The master thesis claims that numbers beyond 10^20 are not feasible, pointing to an asymptomatic runtime nowhere near close to the L[1/3,c] range that the number field sieve has. I don't see where the paper would improve on the results of the master thesis, though