[seqfan] Re: Crypto EllipticK & A000984

[Andrew N W Hone]

Dear Brad, 

Your questions and comments are very worthwhile, in particular 

***** Why should the continuous flow calculated as time evolution coincide
with discrete flows provided by iterating the addition rule?

In general, there is no "natural" discrete version of a Hamiltonian flow. Most discretization schemes (like Euler's method, Newton-Raphson etc.) 
will not give anything nice, even if the original flow is completely integrable. In particular, most discretizations will not preserve the Hamiltonian (energy).

However, for a Hamiltonian system with one degree of freedom (one position x, one momentum y), the level sets of the Hamiltonian H are one-dimensional 
(curves) given by the equation H(x,y)=constant. So if you have a map of the (x,y) plane to itself which also preserves H, then the iterates of this map must lie on the same level curves, 
and so must interpolate the continuous flow. There is a slight subtlety here, which would invalidate what I said: the real level sets of H may have more than one connected component, 
and the map could jump between different components. 

Now the curves 

a=x^2+y^2 +x^2*y^2    

are genus 1, and as a complex curve (Riemann surface) they are connected and smooth for almost all a. A birational map in (x,y) that preserves this curve must be a combination of an involution 
and a translation (this is a general result on automorphisms of elliptic curves, as in e.g. Silverman's book "The Arithmetic of Elliptic Curves"). If the map is orientation-preserving then it is just a translation. 
One can use the parameter t from the Hamiltonian flow as a local complex parameter on the curve, and then one finds that the map is just a shift t -> t + constant. 

Best wishes
Andy 

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Brad Klee [bradklee at gmail.com]
Sent: 25 July 2017 23:05
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Crypto EllipticK & A000984

Hi Andrew and All,

Yes, what you say is also correct. For those who don't already know,
consider that

Area = S(a) = -int x(a,y)*dy = int y(x,a)*dx,

and taking the derivative with respect to "a"

dS(a)/da = int dy/ (-da/dx) = int dx/(da/dy) .

If we have dx/dt = da/dy = (\partial a / \partial y) and dy/dt = -da/dx =
-(\partial a / \partial y), then dS(a)/da, up to a sign, gives an integral
"smooth flow", also called the /time evolution/ in physics. Then the
Hamilton equations provide a third means,

dphi/dt = cos(phi)^2 (d tan(phi)/dt) = (x*(dy/dt) - y*(dx/dt))/(x^2+y^2)

This can be re-written as a ratio of polynomials in variables (x,y), so
easily transforms to polar coordinates, then to a function of phi alone.
After some work on simplification, the expression does reduce to a simple
form, this time:

dphi/dt =  2*sqrt(1-4*a*Z^2).

The factor of two is consistent. It reduces to one by choosing different
Hamilton's equations, dy/dt = -(1/2)*(\partial a / \partial x), dx/dt =
(1/2)*(\partial a / \partial y). Conventions determine the overall scale,
so it's usually okay to make whatever choice sets the period function equal
to 1 in the limit of small amplitude.

I don't think this demonstration of consistency makes my earlier remark any
more inane. Merely stating derivatives does not bring to the fore a
connection between differential equations and addition rules.

***** Why should the continuous flow calculated as time evolution coincide
with discrete flows provided by iterating the addition rule?

I will continue to struggle with Edward's original paper and check the
reference (thank you) to see if it contains any explanation I can
understand. Until then, let's assume ( perhaps incorrectly ) that there
exists a higher perspective where it is obvious that the continuous time
evolution of an integrable Hamiltonian system determines a discrete map
(Q^2,Q^2) |---> Q^2 . Consequently it would become possible to explore
ciphers based on algebraic curves, a superset of elliptic curves.

For example, taking the Hamiltonian form of Edward's curve as a map from
the unit disk to the unit square, we generalize to a polynomial that maps a
unit disk to a unit hexagon:

A - 9*A^2 + 12*A^3 + 12*A^3*X = a / 36 = b ;
where 18*A = r^2 and X = cos(6*phi) transform to polar coordinates (r,phi).

Under these conventions the critical points, vertices of a regular hexagon,
are located at a,r=1; phi = pi/2+n*pi/3. Now A can be solved as a function
of (phi,a) or (phi,b) exactly or via series reversion. Calculating "time
evolution", the period function can be written:

*as coefficients of powers of b, offset 0:
T = (1/2/pi)*int_0^{2pi} dphi (dA/db) =
1, 18, 450, 12420, 360450, 10797948, 330862644 . . .

* as coefficients of powers of a, offset 0:
T = 36*(1/2/pi)*int_0^{2pi} dphi (dA/da) =
1, 1/2, 25/72, 115/432, 2225/10368, 3703/20736, 1021181/6718464 . . .

numerators: 1,1,25,115,2225,3703,1021181 . . .
denominators: 1,2,72,432,10368,20736,6718464 . . .

I have yet to find either of these expansions in OEIS, but who knows? At
least it isn't EllipticK.

Finally, is there some discrete, rational addition system on a
hexagonal-symmetric, 2pi-periodic curve, that looks like continuous time
evolution in the limit of a small increment? I wonder on this point, yet I
do not know...

Thanks,

Brad

Errata: in a previous email, I write "a = x^2 + y^2 + x^2*y^2" rather than "a
= x^2 + y^2 - x^2*y^2". This is a typo; though, mostly inconsequential. And
divise should rather be devise, apologies.

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