Discussion:
Introduction to ECC
(too old to reply)
Jan Dušátko
2018-03-01 07:36:02 UTC
Permalink
Dear,

may someone will be interested, may someone can comments work of my friend

Introduction to ECC - Riemann-Roch transformation
https://trustica.cz/2018/02/22/introduction-to-elliptic-curves/

Introduction to ECC - Elliptic Curves over finite fields
https://trustica.cz/2018/03/01/elliptic-curves-over-finite-fields/

Regards

Jan
Brad Klee
2018-03-01 17:03:46 UTC
Permalink
Hi Jan,

I think you probably want to hear from an expert, but maybe you will find
worthwhile a few comments from another perspective:

1. If you even mention magic in your business pitch-- this is an
advertisement, right?--your potential customers will wonder if you really
have the most powerful magicians on your team. If no, then some potential
customers will run away from you.

2. Mentioning magic also raises suspicion. What assurance do we have that
your suggested website is nothing mendacious? To make the trust situation
more difficult, we can see that trustica is affiliated with brmlab, a
self-declared "hacker" entity based in Prague [1,2].

These points aside, a few more:

3. The supremacy of particular normal form is part of a history leading up
to the computer age, and not really part of the computer age itself. With
computer-based mathematics, we can rely less and less upon simplifying
assumptions. Why do we need to start with Weierstrass formalism?

4. If you look at the [curves] archives, many of the recent topics regard
Curve25519 and Ed25519. Why not use this curve as an example? Can you make
a nice transformation video between these curves?

5. Your mention of Riemann-Roch is confusing. When discussing genus, it's
really more practical and helpful to just give readers the simple Singular
algorithm for calculating the genus of an algebraic plane curve [3] (thanks
again TU Eindhoven!).

4. The citations are superfluous and lacking. LMFDB is also standardized on
the Weierstrass model, so why not cite them? If you search their collection
of Q-Curves for "[2,2,-1,1,3]" or "[-3,21/4]" they will return
"y^2+y=x^3-3*x+5" and "P=(1/4,13/8)" [4].

Regards,

Brad

[1] https://myip.ms/view/dns/571223/ns.trustica.cz
[2] https://wiki.hackerspaces.org/Brmlab
[3] https://www.win.tue.nl/~aeb/2WF02/genus.pdf
[4] http://www.lmfdb.org/EllipticCurve/Q/10179/b/1

Original Post: https://moderncrypto.org/mail-archive/curves/2018/000975.html
Cecilia Tanaka
2018-03-13 00:26:38 UTC
Permalink
On Mar 1, 2018 14:03, "Brad Klee" <***@gmail.com> wrote:

<SNIP>


2. [...] To make the trust situation more difficult, we can see that
trustica is affiliated with brmlab, a self-declared "hacker" entity based
in Prague [1,2].


<SNIP>


[1] https://myip.ms/view/dns/571223/ns.trustica.cz
[2] https://wiki.hackerspaces.org/Brmlab


<SNIP>

Hi, Brad! Hope you are doing good. :)

Sorry for asking something so obvious, but do you really know what is a
hackerspace? :-/

Pardon, but your reference to 'brmlab' was not gentle or even polite, and
every single person who _really_ knows what is a hackerspace, its
importance, and what it means to hackers in the whole world, also knows how
important is being excellent to each other.

I sincerely thank you for discussing the posts and their subjects, but your
reference to 'a self-declared "hacker" entity' was very disgusting.
Please, search a bit more about hackerspaces, makerspaces and fablabs,
about their importance in poor countries like mine, for example.

I live in Sao Paulo, Brazil, and because of a hackerspace, I became a
lawyer who thinks that a simple LED blinking or a little bit of coding can
bring happiness to poor kids and change their lives. When you use a
hackerspace resources for teaching, for example, how to get more political
transparency - wow! - it's pure, astonishing beauty, I swear! <3

So, I sincerely ask you, *please respect hackerspaces, makerspaces and
fablabs*. Everyone deserves respect and it's pretty unfair judging any
kind of space without knowing its use.

Wish lovely nights and days to you and all the Curves members! <3

Ceci... not a 'brmlab' member, but always a hackerspaces lover, yay! ;D
----------
"Don't let anyone rob you of your imagination, your creativity, or your
curiosity. It's your place in the world; it's your life. Go on and do all
you can with it, and make it the life you want to live." - Mae Jemison
Brad Klee
2018-03-13 02:40:41 UTC
Permalink
Hi Cecilia,

Great post, and you are right about hacker hysteria. Maybe you miss my
sense of humour though. It's certainly //not an attack// of hacker culture
that serves the children. I hope that students in Brazil will do crazy
experiments like tearing apart a computer mouse to accomplish a digital
pendulum experiment [1,2]. Even better if observing energy-dependence of
the oscillation period somehow leads into a discussion of elliptic curve
cryptography. Do many computer hackers in Brazil know how to produce the
Jacobian Elliptic functions from Edwards's psi function? That is a topic we
are already starting to help with [3]! Before you start a lawsuit against
me, please realize that it will be more difficult for me to give away these
interesting secrets if I'm incarcerated. Hopefully that isn't your goal
here.

Let's avoid silly personal arguments, continue technical discussion, and
stay on the topic of curves, especially over finite fields. Hey look at
this, someone drew Cayley graphs isomorphic by a bi-rational shear
transformation [4,5]. The difficult part is not calculating inverses,
rather dealing with case exceptions and points at infinity. From whatever
I've managed to figure out, calculations on the Jacobi Quartic probably
need projective coordinates(?). Unfortunately, I don't have anyone telling
me the answers, and this trustica introduction only deals with Weierstrass
normal form. What about pairing calculations? Thanks again Craig Costello,
for making these detailed notes freely available [6]. Learning more soon.

Cheers,

Brad

PS. Don't sell Brazil short, a great country, especially for music, but
they never should have put Caetano Veloso in jail [7]!

[1] https://arxiv.org/pdf/0901.4319.pdf
[2] https://arxiv.org/pdf/1605.09102.pdf ( due for a rewrite )
[3] http://demonstrations.wolfram.com/EdwardssSolutionOfPendulumOscillation/
[4] https://moderncrypto.org/mail-archive/curves/2017/000951.html
[5] Loading Image...
[6] http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf
[7] Caetano Veloso, "It's a long way", Transa, 1972.
Brad Klee
2018-03-05 21:36:52 UTC
Permalink
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,

Brad

after: https://moderncrypto.org/mail-archive/curves/2018/000975.html
[1]

[2] Loading Image...
[3]

[4]

Dominik Pantůček
2018-03-13 08:30:28 UTC
Permalink
Hi Brad and others,

I finally subscribed to the curves list (as Jan has suggested for a long
time) and I think I can give you some insight into GF(p) visualizations
and the behaviour or most (if not all) elliptic curves over finite fields.
Post by Brad Klee
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?
Short reply: you are wrong here.

Medium-sized reply: Needlessly difficult question is finding the set of
right coefficients j1,k1.

Long reply would probably go along the following lines:

Consider an elliptic curve E over the set of real numbers R. That is a
set of points satisfying the equation f(x,y)=0 where f(x,y) represents
the EC equation with all terms on the left side and remaining zero on
the right side.

For simple Weierstrass form that means: f(x,y)=y^2-x^3-ax-b

Now if we consider the same elliptic curve over a prime finite field
GF(p), we get f(x,y)\equiv 0\mod p

(I assume, you are used to TeX notation here - if not, I can produce a
short PDF with everything).

This can be translated to f(x,y)\mod p=0\mod p

As 0\mod p=0, we can simplify it as f(x,y)\mod p=0 (someone might
consider this inaccurate notation, but treating integer division
remainder as operation is more intuitive)

Now, you are absolutely right that f(x+mp,y+np)\mod p=0; x,y\in N+{0};
m,n\in Z is the set of curves containing the solutions which - if
plotted with x,y\in R\and x,y\in<0,p) -- cross the rational points of GF(p).

This is the crucial part - only for some coefficients m,n\in N+{0} you
cross the rational points of GF(p) projected over R.

The visualization of GF(p) on torus contains all m,n\in Z -  beware, for
some functions you get periodic repetitions: for example in my latest
article (which Jan had already posted here) you can see f(x): y=ax+b;
that is - a straight line. And of course it becomes closed over the
torus representing GF(p) (and in this case it hits exactly p number of
points of given GF(p)).

So yes, any particular curve f(x,y)=0 where x,y\in R plotted over GF(p)
region of the real plane will always cross all the rational points of
the same curve f(x,y)\equiv 0\mod p - that is over GF(p).

Yes, certain repetitions (actually infinitely many of them) cross no
rational point at all! That is fine. The field contains only a finite
number of elements and the curve over R is infinite.
Post by Brad Klee
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.
Firstly, the critique is (again) welcome. And secondly... if I read the
rest correctly, you are dealing with ECs over C^2, that is technically
over quaternians... which means your work starts in 4D. As of now I have
no idea how to start thinking about creating decent visualizations of
such. I am really looking forward seeing anything done in that matter.

My interest (for now) lies in ECs over R and GF(p) with emphasis of
creating visualizations and animations of all operations on simple
Weierstrass form, Montgomery form and (yes) Twisted Edwards. The last
one would be really tough on the torus, as you need to project a
hyperbole there.
Post by Brad Klee
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!
It does not look bad for a start. How do you specify the curves and how
do you plot them? For anything smooth I ended up with a naive algorithm
that can plot anything in the form f(x,y)=0 over R and then I just use
coordinate transformations to get the desired pictures.


Cheers,

Dominik
Post by Brad Klee
Again, I'm not an expert, so corrections are welcome if I happen to be
in error.
Cheers,
Brad
after: https://moderncrypto.org/mail-archive/curves/2018/000975.html
[1] http://youtu.be/mFVKuFZ29Fc
[2] https://ptpb.pw/1WvB.png
[3] http://youtu.be/X5hkRIk81_Q
[4] http://youtu.be/zxQSB8OrlZg
Brad Klee
2018-03-13 20:07:03 UTC
Permalink
After: https://moderncrypto.org/mail-archive/curves/2018/000982.html
And: https://moderncrypto.org/mail-archive/curves/2018/000981.html

Hi Dominik,

I am not wrong. In this disagreement, you seem confused about solutions of
diophantine equations. You suggest that a curve will have as many distinct
solutions over Z^2 as over (Z/pZ)^2 with prime p. This counters common
sense, which states that there are not arbitrarily many solutions to any
particular diophantine equation, i.e. over Z^2.

So what is actually happening?

Take the example curve: 0 = -36 + 400*x^2 - 2000*x^2 y + 400*y^2 depicted
over a finite field in the left panel of [1]. This is another form of the
Jacobi quartic, but we might consider it a "new curve" because the
addition rule on the cubic involves a linear intersection geometry,
different from the quartic. More heresy, the x=0 solution occurs at y=2.
Starting with P =[2/5,7/10], we calculate sequences over Q and GF[23],

nP : [2/5, 7/10], [-(7/60), -(11/45)], [-(110/527), 3367/9610],
[184223/336840, 125521/804005], [172557142/149756395, 546265447/84739210] .
. .
nP%23 : [5, 3], [11, 11], [9, 15], [17, 6], [4, 18], [20, 10], [21, 20],
[0,2], [2, 20], [3, 10], [19, 18], [6, 6], [14, 15], [12, 11], [18,3], [
infty,0 ] . . . Repeats . . .

This example shows what actually happens. The Q-sequence visits infinitely
many //fractions// of increasing complexity. Introducing a finite field
such as GF[23], we find the subgroup structure used in ECC. Please realize
that the map from Q^2 ---> (Z/pZ)^2 //is not// a modulus-type map!

The other issue is evaluation of nP using the quartic rule. Apply shear
transform P=[2/5,7/10] ---> P' = [2/5, 3/10]. Then calculate via EFD
formulas:

nP' : [2/5, 3/10], [-(36/35), 1203/490], [-(110/527),
670563/2777290], [202104/921115, 80558112963/339381137290] . . .
nP' % 23 : [5, 21], [20, 1], [9, 8], [17, 15], [4, 1], [11, 4], [21,10], [
infty,0], [2, 10], [12, 4], [19, 1], [6, 15], [14, 8], [3, 1], [18, 21],
[0, 21] . . . Repeats . . .
nP % 23 : [5, 3], [20, 12], [9, 15], [17, 13], [4, 18], [11, 19], [21,
20], [infty, 0], [2, 20], [12, 19], [19, 18], [6, 13], [14, 15], [3,12],
[18, 3], [0, 21] . . . Repeats . . .

Now compare cosets C_16/C_2. They are the same between iterations. The two
C_8 subgroups are different:

G1: [11, 11], [17, 6], [20, 10], [0, 2], [3, 10], [6, 6], [12, 11], [Infty,
0]
G2: [20, 12], [17, 13], [11, 19], [infty, 0], [12, 19], [6, 13], [3, 12],
[0, 21]

Maybe G1 and G2 such as these could lead to an interesting pairing scheme
[2] ? More on this question and evaluation over C & C^2 soon.

Cheers,

Brad


[1] https://ptpb.pw/AfTT.png
[2] http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf
Dominik Pantůček
2018-03-13 21:34:21 UTC
Permalink
Hi Brad,

I will try to keep this short.
Post by Brad Klee
After: https://moderncrypto.org/mail-archive/curves/2018/000982.html
And: https://moderncrypto.org/mail-archive/curves/2018/000981.html
Hi Dominik,
I am not wrong. In this disagreement, you seem confused about
solutions of diophantine equations. You suggest that a curve will have
as many distinct solutions over Z^2 as over (Z/pZ)^2 with prime p.
This counters common sense, which states that there are not
arbitrarily many solutions to any particular diophantine equation,
i.e. over Z^2.
No, I am stating that all solutions over R^2 mapped as repetitions with
coordinate translation x'=x+mp, y=y+np for all m,n\in Z contain all
curve solutions over (Z/pZ)^2. That means, that "infinitely wrapping a
smooth curve over a torus" is the right projection if we want to depict
the repetitive nature of GF(p) in two dimensions (circle would work in
one dimension) for any given curve f(x,y)=0.
Post by Brad Klee
So what is actually happening?
Take the example curve: 0 = -36 + 400*x^2 - 2000*x^2 y + 400*y^2
depicted over a finite field in the left panel of [1]. This is another
form of the Jacobi quartic, but we might consider it  a "new curve"
because the addition rule on the cubic involves a linear intersection
geometry, different from the quartic. More heresy, the x=0 solution
occurs at y=2. Starting with P =[2/5,7/10], we calculate sequences
over Q and GF[23],
Assuming f(x,y)= -36 + 400*x^2 - 2000*x^2 y + 400*y^2 we see that there
are smooth curves drawn in the form of f(x+mp,y+np)=0 for some chosen
values of m and n. And of course, the values are chosen in a way that
shows only parts of infinitely-wrapped smooth curve (or a distinct
number of "new curves" with aforementioned coordinate translation) that
actually hit some interesting points in the (Z/pZ)^2 domain. That is
fine - of course you cannot work with the rest of the points I always
draw and I am talking about. They are there just to help the reader
think about the very nature of the underlying field.
Post by Brad Klee
nP        : [2/5, 7/10], [-(7/60), -(11/45)], [-(110/527), 3367/9610],
[184223/336840, 125521/804005], [172557142/149756395,
546265447/84739210] . . .
nP%23 : [5, 3], [11, 11], [9, 15], [17, 6], [4, 18], [20, 10], [21,
20], [0,2], [2, 20], [3, 10], [19, 18], [6, 6], [14, 15], [12, 11],
[18,3], [ infty,0 ] . . . Repeats . . .
This example shows what actually happens. The Q-sequence visits
infinitely many //fractions// of increasing complexity. Introducing a
finite field such as GF[23], we find the subgroup structure used in
ECC. Please realize that the map from Q^2 ---> (Z/pZ)^2 //is not// a
modulus-type map!  
Ok, I am not going into big detail here, but let me show you, that it
may be a modulus-type map after all.

\frac{2}{5}\equiv 2\frac{1}{5}\mod 23

So we find a multiplicative inverse of 5 over GF(23), which is 14 as
14\cdot 5\equiv 1\mod 23

Therefore \frac{2}{5}\equiv 2\cdot 14\mod 23, which gives us
\frac{2}{5}\equiv 28\mod 23 which yields (unsurprisingly) 5.

The same applies for 7/10 which translates to 3 using the same modular
arithmetic.

We surely confirm that both f(5,3)\equiv 0\mod 23 as it is -136436\mod
23=0, what about the rational notation then? It actually is the case
that f(2/5,7/10)=0 so both solutions are apparently valid over GF(23).

Now let's find f(5+23m,3+23n) to prove that my hypothesis might be
right. If we are unable to find m,n\in Z, I am wrong, if we can find it,
I might be right after all.

And for example f(5+23,3)\equiv 0\mod 23 (looks like infinitely many
repetitions hit this point actually).

I think I understand your point, but that does not invalidate my
hypothesis (which I think can be really easily proven, because if there
exist particular solutions in m and n, those solutions are present in
the set of all m,n\in Z). So again, the curve depicted in the left panel
of [1] can be drawn over a area of R^2 which covers the (Z/pZ)^2 domain
and if we select only those points that fall into (Z/pZ)^2 from the
infinite wrapping around this part of R^2, they do indeed contain all
solutions of such curve over given (Z/pZ)^2 - that is GF(23) in our case.

Please mind that I am primarily talking about finding rational points
over GF(p) now. You are slightly mixing it with the group addition law
used with given curve. Those are not the same things, although you may
typically use the group addition law to generate the rational points
over GF(p) - but sometimes not all. Take Ed25519 for example. You have
to chose generators from all subgroups in order to generate all points
of Ed25519 this way. This is why you need to zero out specified bits
when you are using Ed25519 as a building block of some cryptosystem
(typically for signatures here). If you fall into a small subgroup, you
are screwed in case of signatures. In case of finding all rational
points, you just do not find them all.

I didn't study the addition law used with the curve you posted (but I
have put it on my TODO list) - but if you are saying that it involves
linear intersections, I can also state that such linear intersection
rule would also work by wrapping the line infinitely around the torus
and it would eventually hit the given rational point which maps to the
same rational point over Q you mention above. This is no coincidence,
but again it is a feature of the underlying field. I hope I can show
this in the next video (it is getting tricky). And to add more
confusion, for every line over R^2 (or Q^2 thereof) you can draw two
different lines on the torus having the same properties when it comes to
explaining the group law. It is not in my writing plan (yet), but it can
be really easily shown in Weierstrass simple form (and that is another
reason I started with creating visualizations using the Weierstrass
simple form - it really helps you develop the necessary tools).
Post by Brad Klee
The other issue is evaluation of nP using the quartic rule. Apply
shear transform P=[2/5,7/10] ---> P' = [2/5, 3/10]. Then calculate via
nP'          : [2/5, 3/10], [-(36/35), 1203/490], [-(110/527),
670563/2777290], [202104/921115, 80558112963/339381137290] . . .
nP' % 23 : [5, 21], [20, 1], [9, 8], [17, 15], [4, 1], [11, 4],
[21,10], [ infty,0], [2, 10], [12, 4], [19, 1], [6, 15], [14, 8], [3,
1], [18, 21], [0, 21] . . . Repeats . . .
nP % 23  : [5, 3], [20, 12], [9, 15], [17, 13], [4, 18], [11, 19],
[21, 20], [infty, 0], [2, 20], [12, 19], [19, 18], [6, 13], [14, 15],
[3,12], [18, 3], [0, 21] . . . Repeats . . .
Now compare cosets C_16/C_2. They are the same between iterations. The
G1: [11, 11], [17, 6], [20, 10], [0, 2], [3, 10], [6, 6], [12, 11],
[Infty, 0]
G2: [20, 12], [17, 13], [11, 19], [infty, 0], [12, 19], [6, 13], [3,
12], [0, 21]
Maybe G1 and G2 such as these could lead to an interesting pairing
scheme [2] ? More on this question and evaluation over C & C^2 soon.
Now we are talking about something completely different. The generated
subgroups are an interesting feature of given curve and it is nice that
it can be shown on GF(p) with reasonably small p. About the pairing -
well ... I would love to read about that if you find anything
interesting there. Right now I am knee-deep in Ed25519 and Curve25519
embedded implementations and making nice pictures and videos makes me
breath easier :)

And I am really curious about your work over C^2 here. Just writing
about that makes me want to get back to academia :)


Cheers and keep up the good work!
Dominik

P.S.: It wasn't short after all ... my apologies for that.
Post by Brad Klee
 
Cheers,
Brad
[1] https://ptpb.pw/AfTT.png
[2] http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf
Brad Klee
2018-03-14 00:02:41 UTC
Permalink
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!

Brad

Jan Dušátko
2018-03-12 11:20:08 UTC
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Post by Jan Dušátko
Dear,
may someone will be interested, may someone can comments work of my friend
Introduction to ECC - Riemann-Roch transformation
https://trustica.cz/2018/02/22/introduction-to-elliptic-curves/
Introduction to ECC - Elliptic Curves over finite fields
https://trustica.cz/2018/03/01/elliptic-curves-over-finite-fields/
3rd part of Introduction to ECC -  Point negation

https://trustica.cz/2018/03/08/elliptic-curves-point-negation/

Regards

Jan
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