[seqfan] Re: Crypto EllipticK & A000984

Brad Klee bradklee at gmail.com
Wed Jul 26 00:05:00 CEST 2017

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...



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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