Point addition / Point concatenate

[brainless]

abc
023ef30130654689a64c864d6dd38760481c55fc525e2c6c7084e2d2d3d4d51be9
043ef30130654689a64c864d6dd38760481c55fc525e2c6c7084e2d2d3d4d51be9f7d86b288c09d db5311f292285168000e43e4b62201bd8de23a391daa8e00ce8

def
036a6e1dc6f203f7fdd97965892301e5fb995a37318c410543835f0edcd3456c49
046a6e1dc6f203f7fdd97965892301e5fb995a37318c410543835f0edcd3456c492a072b9898b93 e9eb05f9ad86a97546d83b579bf6efd3482f93baca13784496b

abcdef
0312faae608bd6562562b8f85564664cd1fdcd667f6b24b2b221ef86b9231f4d74
0412faae608bd6562562b8f85564664cd1fdcd667f6b24b2b221ef86b9231f4d74512ee8cd9b343 31afd05ccb8d81d1393c150c73ec5695845b731f7e6e0086719

here abc is 1 point, and 2nd point is def
if we can join or merge like abc add to def
where result would be abcdef point

abc+def = 18AB
one new func could be inside
like abc+def = abcdef
like point +point = pp ( line extended) ( not 2p)

point 1
point 2
a =1+2 = 3
more advance looking func

point 1
point 2
a =1+2 = 12

any ecc expert, can explain about this addition apply?
maybe it will called Point concatenate

[NotATether]

See this thread. I quote a post from there:

To get C = A + B, where A != B  (in other words A and B must not be the same point. If that's the case, use point doubling instead):

- lambda = (By - Ay)/(Bx - Ax)   (Note: no a term, even if the curve is not secp256k1!)
- Cx = lambda² - Ax - Bx
- Cy = lambda*(Ax - Cx) - Ay

So point addition isn't concatenation or arithmetic addition, you simply take their uncompressed form, extract the X and Y values and perform the above operations on them and embed the X and Y result into an uncompressed point.

Note: "abc" and "def" don't look like curve points, because they are just hex numbers with no X and Y value.

[MixMAx123]

The points are coded like a public key:

abc A: 3ef30130654689a64c864d6dd38760481c55fc525e2c6c7084e2d2d3d4d51be9
abc B: f7d86b288c09ddb5311f292285168000e43e4b62201bd8de23a391daa8e00ce8

def A: 6a6e1dc6f203f7fdd97965892301e5fb995a37318c410543835f0edcd3456c49
def B: 2a072b9898b93e9eb05f9ad86a97546d83b579bf6efd3482f93baca13784496b


abc + def:
15cb9594ee936bd14b98139460879bee6f33b0f7c0ccb293825794d985f595be
aaad772a2ec81ab6013775e9edb274efe2e133e65e58949ac7f25429fb1bb152

[brainless]

P + G = PG

hex prvkey is abc
x: 3ef30130654689a64c864d6dd38760481c55fc525e2c6c7084e2d2d3d4d51be9
y: f7d86b288c09ddb5311f292285168000e43e4b62201bd8de23a391daa8e00ce8

hex prvkey is def
x: 6a6e1dc6f203f7fdd97965892301e5fb995a37318c410543835f0edcd3456c49
y: 2a072b9898b93e9eb05f9ad86a97546d83b579bf6efd3482f93baca13784496b


result = hex prvkey is abcdef:

x: 12faae608bd6562562b8f85564664cd1fdcd667f6b24b2b221ef86b9231f4d74
y: 512ee8cd9b34331afd05ccb8d81d1393c150c73ec5695845b731f7e6e0086719

[brainless]

https://crypto.stackexchange.com/questions/310/mapping-points-between-elliptic-curves-and-the-integers

is this formula can work on above required result ?

[NotATether]

https://crypto.stackexchange.com/questions/310/mapping-points-between-elliptic-curves-and-the-integers

is this formula can work on above required result ?

The formulas in that post are for compressing an X,Y pair into an x- coordinate, and has nothing to do with point addition. I have already given the formula for that in my first post, and that's what I'm going to assume you used to derive your answer:

x: 12faae608bd6562562b8f85564664cd1fdcd667f6b24b2b221ef86b9231f4d74
y: 512ee8cd9b34331afd05ccb8d81d1393c150c73ec5695845b731f7e6e0086719

[brainless]

https://crypto.stackexchange.com/questions/310/mapping-points-between-elliptic-curves-and-the-integers

is this formula can work on above required result ?

The formulas in that post are for compressing an X,Y pair into an x- coordinate, and has nothing to do with point addition. I have already given the formula for that in my first post, and that's what I'm going to assume you used to derive your answer:

x: 12faae608bd6562562b8f85564664cd1fdcd667f6b24b2b221ef86b9231f4d74
y: 512ee8cd9b34331afd05ccb8d81d1393c150c73ec5695845b731f7e6e0086719

your formula if write in python, will help me to understand, kindly write python script, thankx

[NotATether]

your formula if write in python, will help me to understand, kindly write python script, thankx

Code:

def point_add(Ax, Ay, Bx, By):
    """Point addition of points (Ax, Ay) and (Bx, By)"""
    
    l = (By - Ay)//(Bx - Ax)
    Cx = l**2 - Ax - Bx
    Cy = l*(Ax - Cx) - Ay
    return Cx, Cy   # Point (Cx, Cy) = (Ax, Ay) + (Bx, By)

_{small edit: the code was full of no-break spaces}

Note that you will have to unserialize your hex numbers into integers before you can pass them here. The integer type in Python (unlike in other languages) is infinite-precision.

 File "main.py", line 33
    lambda = (By - Ay)/(Bx - Ax)
           ^
SyntaxError: invalid syntax

Right. lambda is a keyword in Python so I have to name it something else. See the edited code.

[brainless]

your formula if write in python, will help me to understand, kindly write python script, thankx

Code:

def point_add(Ax, Ay, Bx, By):
    """Point addition of points (Ax, Ay) and (Bx, By)"""

    lambda = (By - Ay)/(Bx - Ax)
    Cx = lambda**2 - Ax - Bx
    Cy = lambda*(Ax - Cx) - Ay
    return Cx, Cy   # Point (Cx, Cy) = (Ax, Ay) + (Bx, By)

Note that you will have to unserialize your hex numbers into integers before you can pass them here. The integer type in Python (unlike in other languages) is infinite-precision.

  File "main.py", line 33
    lambda = (By - Ay)/(Bx - Ax)
           ^
SyntaxError: invalid syntax

[brainless]

if
Ax = 28472748655393009267694063518808149386163109418078082138878164433505686461417 #3ef30130654689a64c864d6dd38760481c55fc525e2c6c7084e2d2d3d4d51be9
Ay = 112103652146063681709271542430359369857619412145509508700963842184555907779816 #f7d86b288c09ddb5311f292285168000e43e4b62201bd8de23a391daa8e00ce8

Bx = 48139720641152563685347991770512702742210610129134046538275528729266911276105 #6a6e1dc6f203f7fdd97965892301e5fb995a37318c410543835f0edcd3456c49
By = 19009808459039115948789087270292949097337312255656245874893769987807919753579 #2a072b9898b93e9eb05f9ad86a97546d83b579bf6efd3482f93baca13784496b

what result will be back from your formula ?
x: 9858272149801355672780957992190489068105775500642985285161445958802893804990
y: 77199671246421658177371783275117138254582110578372826724463426751083822428498

or

x: 8584546547439194144767466770313806210515315127328165539072185327841223593332
y: 36720222448510784205200470825713773455378994828752011866054770511274779895577

[o_e_l_e_o]

-snip-

If your two points are:

Code:

043EF30130654689A64C864D6DD38760481C55FC525E2C6C7084E2D2D3D4D51BE9F7D86B288C09DDB5311F292285168000E43E4B62201BD8DE23A391DAA8E00CE8
046A6E1DC6F203F7FDD97965892301E5FB995A37318C410543835F0EDCD3456C492A072B9898B93E9EB05F9AD86A97546D83B579BF6EFD3482F93BACA13784496B

Then adding them together gives you the following point:

Code:

0415CB9594EE936BD14B98139460879BEE6F33B0F7C0CCB293825794D985F595BEAAAD772A2EC81AB6013775E9EDB274EFE2E133E65E58949AC7F25429FB1BB152

If you want to break that in to an x and y coordinate and convert to decimal as you have done, then the coordinates are:

Code:

x = 9858272149801355672780957992190489068105775500642985285161445958802893804990
y = 77199671246421658177371783275117138254582110578372826724463426751083822428498

[brainless]

-snip-

If your two points are:

Code:

043EF30130654689A64C864D6DD38760481C55FC525E2C6C7084E2D2D3D4D51BE9F7D86B288C09DDB5311F292285168000E43E4B62201BD8DE23A391DAA8E00CE8
046A6E1DC6F203F7FDD97965892301E5FB995A37318C410543835F0EDCD3456C492A072B9898B93E9EB05F9AD86A97546D83B579BF6EFD3482F93BACA13784496B

Then adding them together gives you the following point:

Code:

0415CB9594EE936BD14B98139460879BEE6F33B0F7C0CCB293825794D985F595BEAAAD772A2EC81AB6013775E9EDB274EFE2E133E65E58949AC7F25429FB1BB152

If you want to break that in to an x and y coordinate and convert to decimal as you have done, then the coordinates are:

Code:

x = 9858272149801355672780957992190489068105775500642985285161445958802893804990
y = 77199671246421658177371783275117138254582110578372826724463426751083822428498

if we convert back into hexprvkey as per your addition
abc
def
abc+def = 18AB

but what i am looking addition like example in hex prvkeys based on x,y points
abc
def
abc+def = abcdef

[o_e_l_e_o]

but what i am looking addition like example in hex prvkeys based on x,y points
abc
def
abc+def = abcdef

Well first, these aren't private keys but rather points on the curve, which would be public keys if anything.

Yes, I guess you could concatenate two points together. You could take the two x coordinates:

Code:

3EF30130654689A64C864D6DD38760481C55FC525E2C6C7084E2D2D3D4D51BE9
6A6E1DC6F203F7FDD97965892301E5FB995A37318C410543835F0EDCD3456C49

Concatenate them together:

Code:

3EF30130654689A64C864D6DD38760481C55FC525E2C6C7084E2D2D3D4D51BE96A6E1DC6F203F7FDD97965892301E5FB995A37318C410543835F0EDCD3456C49

Modulo p:*

Code:

0D1C311FC0BD9915B98E45F586FE55821BAE99FB78ADB182BCCDC84652001073

And then calculate the two corresponding y coordinates:

Code:

76C85E288DE490C4281D4E5C94F930900FABA66B47B4CF1788C657BAF15C9BBA
8937A1D7721B6F3BD7E2B1A36B06CF6FF0545994B84B30E87739A8440EA36075

I have no idea why you would want to do such a thing though. Care to explain what you are trying to achieve here?


*p is the finite field over which the secp256k1 curve is defined:

Code:

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F

[brainless]

here http://web.math.princeton.edu/swim/SWIM%202010/Yao-Zhan%20Presentation%20SWIM%202010.pdf
page 5
Adding points is not the same addition as
1+1=2.
this line make me thinking
if its not 1+1 = 2
then could be 1+1 = 11
just this make me thinking different and test

[o_e_l_e_o]

if its not 1+1 = 2
then could be 1+1 = 11

No, it is neither 1+1 = 2 nor is it 1+1 = 11. The correct answer has been given by NotATether in the first reply in this thread.

Graphically, you plot the two points on the curve, draw a straight line between them, extend that straight line to find the third point where it intersects the curve, and then reflect that third point across the x axis to find your answer.

Algebraically, you use the equations given above.

This is explained on page 8 of the link you gave, with the (slightly more complicated) equations given on page 10.

[mrxtraf]

Here you write 1 + 1 = 11
And in what system should this be applied? Decimal, hex, or maybe binary?
For example, two numbers in hex b(11) and d(13)
In hex
b + d = bd
But in dec
11 + 13 = 1113
But
hex bd = dec 189 and not equal 1113 from previso result
So which system should you use as a basis for concatenation?
If decimal
X + Y = XY
in mathematic

X*strpad('1', len(Y)+1, '0', right) + Y

But in Hex it is completely different!

But if we transfer this to the points of coordinates, then nothing will work, because the length of Y is not known.

[brainless]

Here you write 1 + 1 = 11
And in what system should this be applied? Decimal, hex, or maybe binary?
For example, two numbers in hex b(11) and d(13)
In hex
b + d = bd
But in dec
11 + 13 = 1113
But
hex bd = dec 189 and not equal 1113 from previso result
So which system should you use as a basis for concatenation?
If decimal
X + Y = XY
in mathematic

X*strpad('1', len(Y)+1, '0', right) + Y

But in Hex it is completely different!

But if we transfer this to the points of coordinates, then nothing will work, because the length of Y is not known.

here we are taking as hex
b+d =bd
where b is one point and d is 2nd point, result should be bd point in concatenation
for your easy dec result will be 189 not 1113

[brainless]

hex b dec 11
x: 774ae7f858a9411e5ef4246b70c65aac5649980be5c17891bbec17895da008cb
y: d984a032eb6b5e190243dd56d7b7b365372db1e2dff9d6a8301d74c9c953c61b

hex d dec 13

x: f28773c2d975288bc7d1d205c3748651b075fbc6610e58cddeeddf8f19405aa8
y: 0ab0902e8d880a89758212eb65cdaf473a1a06da521fa91f29b5cb52db03ed81

result should be
hex bd dec 189
x: 6d7ef6b17543f8373c573f44e1f389835d89bcbc6062ced36c82df83b8fae859
y: cd450ec335438986dfefa10c57fea9bcc521a0959b2d80bbf74b190dca712d10

[mrxtraf]

Impossible, not real and not logical.

[brainless]

Impossible, not real and not logical.

movie name mission impossible, always done at possible, i believe some math expert have solution,