[curves] Introduction to ECC

Brad Klee bradklee at gmail.com
Tue Mar 13 17:02:41 PDT 2018

Hi Dominik,

Chalk it up to different conventions. I am thinking of a typical modulus
function for which the input is of the same type as the output; reals to
reals, rationals to rationals, complex to complex, (Z/pZ) to (Z/pZ), etc.
Then: Mod[2/5,23] = 2/5. There is another function QtoF[q,p] which takes
input rational q and finds the correct element from (Z/pZ). This function
QtoF composes three or four elemental functions, including the typical Mod
function and an arithmetic table.

We can agree that lines wrap the torus nicely but cubic or quartic curves
do not. Given three points on the same line, we can agree that these points
may also be points on one elliptic curve up to a composition of Mod and
QtoGF. My contention is that: If you want to represent the solutions as an
intersection geometry on a torus over (Z/pZ)^2, in general you will need
three elliptic curves f1, f2, f3 and two integers (n,m) related by f1 +n*p
= f2 + m*p = f3.

A torus drawing such as you propose could be nice if you are careful with
this issue about Mod & QtoGF. The lines do wrap nicely. However you may
encounter interdisciplinary difficulties with complex people who define
elliptic curve tori in terms of the time-parameterization problem for
reducing C^2 ----> C .

Thanks for your interest, I will remember to send you a copy of my current
writing effort when its finally finished. In these times interdisciplinary
efforts do not happen quickly, especially as math, physics and computer
science are all subjects with different languages and purposes. For the
sake of developing perspective, it's nice to talk to another enthusiast who
is involved with actual details of implementation. Back to writing . . .

Cheers, you too!

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