Topic: I created smaller secp256k1 just for testing (Read 170 times)

The smallest valid example I can come up with:

Curve p=7, n=13, y^2=x^3+3, generator=(1,2)
Server to execute code: https://sagecell.sagemath.org/
Table of private and public keys:
And then, everything can be rotated just like in a clock, with additional 13th point at infinity.

Later, you can try to go into bigger and bigger curves, and get for example everything below 100:
It is just slightly modified example from vjudeu's code: https://github.com/vjudeu/curves1000/wiki/Six-n%E2%80%90values-when-changing-b%E2%80%90value

As you can probably see, you can get these results:
Another similar topic, which may be useful: https://bitcointalksearch.org/topic/smaller-elliptic-curves-y2-x37-based-on-secp256k1-5459153

Here, Garlo Nicon was trying just y^2=x^3+7, and checked only values, where p-value and n-value can be swapped, but as it is not the case in secp192k1 and secp224k1, and works only for secp160k1 and secp256k1, then probably other b-values should be considered as well.

@dexizer7799
Thank you but I think this is an example for different type of curve... I do not know I did not understand the content of that link

for small secp256k1 I found that if you do addition of all X values for all points and do mod p you get 0

So if P=97 I get n=79

if I do the addition of all valid points Gx+2Gx+3Gx+4Gx....+78Gx Where Generator point is P   (1,69) then I get
3880 mod 97 = 0

Maybe this can be usefull

Hola! You can check this one https://amnesia.sh/ctf/2024/03/14/htb-arranged.html

secp256k1 curve is
With the help of cuberoot of Unity, we can say that each value of Y there will be 3 values of X. (Endomorphism Points)
In a similar way For each X there will be 2 values of Y. (symmetry Points)

The relationships between them are well known and already used in various tools for faster scanning (example VanitySearch from JLP).
In a very simple way if you have a Pubkey points (x, y) you can get in total 6 pubkeys easily through

These points are related with the Generator through Privatekeys like this

Here the secp256k1 constants are

Quite impressive!!!
I was doing the same things when i started learning secp256k1 curve.
I do not know if that info can be used to some advantage but there is one good saying: "If one does not know how to do it DOES NOT MEAN THAT IT CANNOT BE DONE!!!"

So I am learning about everything and so there is a chance that this post can be lame/noob or something like that

I created a smaller secp256k1, p=43 n=31
I took point G(2, 31) for the generator point and I got the list of all valid points

I printed this and I put it on my wall so when I work (I work from home) sometimes I look into this paper and I tried to find something interesting (interesting for me )

so this is what I found and also those rules can be applied on regular secp256k1 that Bitcoin is using...

Every Y has 3 different X values... When you calculate G of those points you always get n=31
so G=(2,31) + 5G=(12,31) + 25G=(29,31) = 31G, also for all other points 2G=(7,36) + 10G=(42,36) + 19G=(37,36) = 31G
If you change generator points those positions will remain the same so for n=31 you will always get the same Y on the same sets of points 1 5 25, 2 10 19... there are 10 sets of this points

There are negative points set with 1 5 25 and 6 26 30 (6 26 30 starting from the "end" to start" basically) When you do positive set + negative set of points those point will always return 93G or 3n

I found that if you do for one set of point X1 + X2 +X3 you sometimes get 2P and sometimes get only P I do not know why...

so on original curve where y is 483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
you have 3 X
first   x: c994b69768832bcbff5e9ab39ae8d1d3763bbf1e531bed98fe51de5ee84f50fb
second   x: 79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
third   x: bcace2e99da01887ab0102b696902325872844067f15e98da7bba04400b88fcb

If you do X1 + X2 + X3 in this case you will get 2P in some other cases I found that I only get 1P
Y 4218f20ae6c646b363db68605822fb14264ca8d2587fdd6fbc750d587e76a7ee

first   x: 1
second   x: 7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee
third   x: 851695d49a83f8ef919bb86153cbcb16630fb68aed0a766a3ec693d68e6afa40
In this case you get only P when you do X1 + X2 + X3

I found that for symmetrical points if you do Y1+Y2 you always get P
b7c52588d95c3b9aa25b0403f1eef75702e84bb7597aabe663b82f6f04ef2777 + 79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798 = p
.
.
.

Is there and useful way that we can use those points if we can calculate their G positions? I am testing bigger and bigger curves and always looking for n that number of points mod 6 = 0 (+ infinity point)
p = 937 n = 973
1G 235G 737G

So I am trying to make a connection between 1st and 2nd point for any p and any n....
If I know 1st and 2nd point I can easily calculate 3rd = n - 1st - 2nd
then I can calculate the position of other 3 points that has different Y but same 3 X

If we know the relations between those points maybe we can scan keys faster or see some periodical groups I do not know