[Cryptography] Edwards roller coaster (a physical model)
Bill Cox
waywardgeek at gmail.com
Fri Jun 10 01:04:28 EDT 2016
This is a fun physical model for Edwards curves I came up with, which can
represent most of the elliptic curves we use for crypto that are related to
the Jacobi addition laws
<https://en.wikipedia.org/wiki/Jacobi_elliptic_functions#Addition_theorems>.
I do not know of any relevance this has for crypto, but I like to have a
physical model in my head when thinking about problems...
When d < 0, we get a 4-pointed star inscribed in the unit circle. This
picture from Wikipedia is accurate, except that it flipped the sign of d,
so negate the signs of d and you will be less confused, because then it
will match the article:
[image: Inline image 1]
These squished circles where d < 0 are entirely within the unit circle.
Place a unit hemisphere on top of it, and project the squished circle
directly up onto the hemisphere. Imagine that the resulting path is the
track of a roller coaster. At time t = 0, the roller coaster is moving at
angular velocity w0, crossing the point (x, y, z) = (1, 0, 0). w0 Is
really just the speed at that point, but we need it to have units of
1/seconds for the units to work out.
Now assume that the track is frictionless, and that there is a bizarre
force acting on the roller coaster. For d == -1, the force pushes away
from the Z-axis with strength equal to e^(w0*r), where r is the distance
from the Z-axis, in other words, sqrt(x^2 + y^2). Got it? What happens
when you set off the roller coaster at t = 0, with a speed of w0?
The roller coaster speeds around the track, moving faster near the corners
of the star, and slower as it gets close to the Z-axis. The speed is
always w0/r, and the potential plus kinetic energy is constant. In this
model, the thing being added by the Edwards addition law is time. Given
the position at t1 and t2, we can use the addition law to compute the
position at t1 + t2.
I kind of want to build an animated gif :>
Bill
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