[Cryptography] Why is ECC secure?

[Bill Cox]

On Thu, Aug 13, 2015 at 4:26 PM, Viktor Dukhovni <cryptography at dukhovni.org>
wrote:

>
> Therefore, any proportionality between "n" and the "distance" of
> "nG" from some reference point (pick any continuous metric), fails
> for large enough "n".  Thus, before we even consider whether any
> of this applies to the discrete case, it seems clear that this must
> fail in the continuous case.
>
> --
>         Viktor.
>

The power of visualization seems to be under-rated in group theory :)

Not only does all this work, it gives me a way to create new additive
groups easily, which is something I've wanted to know how to do for a while
now.  For example, here's a group I just came up with using this line
integral stuff:

a @ b  = sqrt(((4a^2 + 1)^(3/2) + (4b^2 + 1)^(3/2))^(2/3) - 1)/2

Plug it into a spread sheet, and you'll see it works.  I created it by
doing the line integral of 12x over the curve y = x^2.  If the line
integral is called L(x), then the addition rule is simply Linv(L(a) +
L(b)).  I'm not sure if the cube-root is friendly modulo a prime, but if it
is, we could probably use this to do crypto :)

We can create groups on path or function using this technique.  The unit
circle group is the simplest case, where point multiplication is simply
angle addition.

Bill
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