From: "Mikael Johansson" <mikael.johansson@wineasy.se>
Subject: Re: transforming elliptic curves (?)
Date: Tue, 13 Jun 2000 21:18:24 +0200
Newsgroups: sci.math
Summary: [missing]

Mike Rosing wrote in message <394674AB.DD84990C@physiology.wisc.edu>...
>Howdy,

Hi

>I've been slowly learning the some of the math associated with the arithmetic
>of elliptic curves.  I have two different "views" of the math - one is from
>the field of cryptography where I work in finite fields, usually in GF(2^n).
>The other is over the complex plane, and I've plotted functions like the
>j invariant and Weierstrass rho(z,tau) using the Fourier expansion forms of
>these functions.
>
>My main question is how to connect these two views.  I know that rho is used
>as a coefficient of an elliptic curve over the complex plane, but I'd like
>to know how to map or transform that into a curve over a specific field in
>GF(2^n) (assuming all parameters are correct for doing that of course).


elliptic curve <-> j-invariant:

Given an elliptic curve in Weierstrass form:
y^2=4x^3-ax-b
the j-invariant is given by
j=1728b^3/(b^3-27a^2)

Thus the j-invariant can be acquired from any elliptic curve.

An elliptic curve giving any specific j-invariant can be found defining the
curve to be

y^2=4x^3-cx-c
and then solving for j in the corresponding equation. If we write j=1728J,
this yields the form
J=c^3/(c^3-27c^2)=c/(c-27)
and thus
c=27J/(J-1)

For J=0, one could use, for instance, y^2=4x^3-3x
and for J=1, for instance y^2=4x^3-1


elliptic curve <-> Weierstrass 'rho':

the rho-function (in most literature denoted by a symbol that's generally
known as 'Weierstrass p'..) or p-function is defined over a lattice
generated by two complex numbers omega_1 and omega_2, or possibly normalized
to but one complex number tau as an infinite sum of certain reciprocals.

The coefficients a and b of the curve are related to the lattice by the
relation given through the expansion of the p-function...

Denote the first few terms of the p-function as

p(z)=1/z^2+3s_4z^2+5s_6z^2+...

where
s_m(omega_1,omega_2)=s_m=sum_(omega =/= 0) 1/omega^m

Then denote

g_2(omega_1,omega_2)=g_2=60s_4
and
g_3(omega_1,omega_2)=g_3=140s_6

This yields
p'^2=4p^3-g_2p-g_3


Given suitable underlying lattice, the coefficients g_2 and g_3 can be made
to take on any value, thus giving a relation between the p-function and any
elliptic curve.

In the same fashion, any given lattice gives a certain elliptic curve.

>If anybody knows how I can do that, I'd appreciate pointers to books.
>Also, you can check out plots of the above functions here:
>http://www.terracom.net/~eresrch/float


I've quite a lot in store, as I approached the same problem, from the
viewpoint of Fermat's last theorem in my extended essay (can be found at
http://haven.myip.org/fermat/ in postscript, pdf or dvi-format)

My posting sofar was written using Serge Lang's 'Elliptic function' as a
reference. Further brilliant books on the subject include
N.I. Akhiezer   'Elements of the Theory of elliptic functions'
B. Shoeneberg   'Elliptic Modular functions'
Dale Husemöller 'Elliptic curves' and the **brilliant**
Neal Koblitz    'Introduction to Elliptic curves and Modular Forms'

>Patience, persistence, truth,
>Dr. mike

Good luck, and I hope I've been comprehensible :)

// Mikael Johansson