# [curves] Introduction to ECC

Mon Mar 5 13:36:52 PST 2018

```Hi Jan and Curve Fans,

After more thought, there is one other nagging issue. Nice quality aside,
I'm not sure that the torus video [1] leads in the right direction.

When dealing with modulus arithmetic, it seems extremely unlikely to me
that one particular curve will visit all valid points. In a field such as
GF(23), we could typically expect to see a range of curves C(x,y)=23*n for
different integer n, as depicted elsewhere [2]. By wrapping the curve
around the torus, your animation seems to say that any particular solution,
C(x1,y1)=23*n1, can be lattice-translated to C(x1+23*j1,y1+23*k1)=0. The
zero-constraint is a Diophantine equation in (j1,k1). I'm not sure if or
how you can guarantee a translation (j,k) exists for every valid triple
(x,y,n)?? Isn't this a needlessly difficult question to answer when you can
just plot a range of curves over one square domain?

Perhaps the critique is overly technical. Even on a qualitative level, the
animation is likely to introduce cognitive dissonance. Your torus domain
goes along two real variables. For doubly-periodic elliptic functions, the
domain covers a portion of the complex plane. Granted, computer science
applications do not really depend on integrals or differential equations.
However, an effort to find the best possible depictions of elliptic curves
should really be interdisciplinary, with ideas and input from all around.

I will mention again that I have yet to see any convincing pictures or
animations of elliptic curves, which emphasize realizations as genus one
surfaces. However, in recent exploration of complex transformation between
families of elliptic curves, I've invented another decent depiction
algorithm [3,4]. Quality is not too bad, though yes, rushed. At least the
drawings have nothing to do with any particular normal form!

Again, I'm not an expert, so corrections are welcome if I happen to be in
error.

Cheers,