[Cryptography] Why is ECC secure?

[Bill Cox]

I took my geometry based attack further today and found some things I think
are very cool.  In particular, in an Edwards curve with negative d
(squished circle, not fat one), I set z = -sqrt(d)xy, so that the Edwards
curve points map onto the unit sphere.  I found that when I add a small
delta (like the point (0.0001, ~0), then measure the distance traversed on
the sphere, it is always equal no matter what point I start from, once I
divide by 1/sqrt(x^2 + y^2).

I computed the line integral on the sphere from the Edwards curve origin
(0, 1, 0) to an arbitrary point using Wolfram's awesome integration
toolkit.  It resulted in a closed form solution, but unfortunately involves
an Elliptic integral.  This was the only part that I can't compute using
modular arithmetic.  Had it resulted in an equation that was modular
arithmetic friendly, I think that might result in a significant break of
elliptic curve crypto that can be mapped to Edwards curves.

The idea would have been to find the modular distance from the origin of
the generator point, and also of the user's publiic key point.  At that
point, I think we've mapped the problem to regular modular arithmetic in
one variable.  But... it didn't work.  Was fun, though :)

I used Wolfram to evaluate the path integral for several multiples of the
generator point, and indeed, they are clearly multiples of a constant.

Bill

Bill
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