Re: I created smaller secp256k1 just for testing

secp256k1 curve is

Code:

y² = x³ + 7

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

Code:

6 Pubkeys = [x,y] [x*beta%p, y] [x*beta2%p, y] [x,p-y] [x*beta%p, p-y] [x*beta2%p, p-y]

These points are related with the Generator through Privatekeys like this

Code:

6 Pvkeys = pvk, pvk*lmda%N, pvk*lmda2%N, N-pvk, N-pvk*lmda%N, N-pvk*lmda2%N

Here the secp256k1 constants are

Code:

p = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
# Constants Based on Cube root of 1
beta = 0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee
beta2 = 0x851695d49a83f8ef919bb86153cbcb16630fb68aed0a766a3ec693d68e6afa40 # beta*beta
lmda = 0x5363ad4cc05c30e0a5261c028812645a122e22ea20816678df02967c1b23bd72
lmda2 = 0xac9c52b33fa3cf1f5ad9e3fd77ed9ba4a880b9fc8ec739c2e0cfc810b51283ce # lmda*lmda