Do we have any algorithm to check for primes in either x or y coordinates?
Just make a little change to the code given by Garlo Nicon, and you will have it:
p=0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
modulo_root=(p+1)/4
x=1
is_on_curve=False
is_running=True
while is_running:
x_cube=(x*x*x)%p
y_square=(x_cube+7)%p
y=y_square.powermod(modulo_root,p)
is_on_curve=(y.powermod(2,p)==y_square)
if is_on_curve and is_prime(x) and is_prime(y):
print(hex(x),hex(y))
is_running=False
x+=1
As you can see, it is easy to get an answer. And if you remove "is_running=False" from the inside of this loop, you will get a lot of points, until you CTRL+C your program, or it will timeout in Sage online server.
0xe9b 0xe22c56c79e9d1ab0357f4348aadb53006efeb69fd3f1924ea0bfe8201d2e1d23
Also do you think there is any use going after p values existing in G and then see if those p values are a valid point on secp256k1 or not?
You can try, but I don't think the solution is there. But well, you can do a lot of things with elliptic curves. Another question is: does it make any sense? For example, here is another game I played some time ago:
True 0x1 0x4218f20ae6c646b363db68605822fb14264ca8d2587fdd6fbc750d587e76a7ee
True 0x4218f20ae6c646b363db68605822fb14264ca8d2587fdd6fbc750d587e76a7ee 0xc8d9659b4430c5c0dd89ce385731acd388dba5e3f24bcbc0e955e2cb602da315
True 0xc8d9659b4430c5c0dd89ce385731acd388dba5e3f24bcbc0e955e2cb602da315 0xa4d87747ecd146a09a14a531b889425c8fe308973b3fce622987d4fe27db17bc
True 0xa4d87747ecd146a09a14a531b889425c8fe308973b3fce622987d4fe27db17bc 0x83a6221ccaf1844405bfd822cd7f99405efc4ad3f458ad08283f5f09a9ade2e2
True 0x83a6221ccaf1844405bfd822cd7f99405efc4ad3f458ad08283f5f09a9ade2e2 0xc9a6eabb6b8f0edec38ab42532d4456b9c76eb2c9cf690f603dcc688566e05d
True 0xc9a6eabb6b8f0edec38ab42532d4456b9c76eb2c9cf690f603dcc688566e05d 0x702d537e9b0d595b72a34e27e2c3a6f0ff3838f30504ce1fd626658cc619c73b
False 0x702d537e9b0d595b72a34e27e2c3a6f0ff3838f30504ce1fd626658cc619c73b 0x420a44c6b6d1fb0fee5f0f533871011470e7fff36bf345c76e300c3862160067
False 0x702d537e9b0d595b72a34e27e2c3a6f0ff3838f30504ce1fd626658cc619c73c 0xb29bb4798b95c80de79a4de56d0c83809c884423c1f81dbe59f06dce8c8b9644
True 0x702d537e9b0d595b72a34e27e2c3a6f0ff3838f30504ce1fd626658cc619c73d 0x5dc0fe834752c56b0402d4adc5db796fb802a5ac2f377148d8270a657541b5f3
But guess what: it is a funny game, but it will give you no solutions to any existing puzzles. And it will not reveal you any private keys, because it works purely on public keys, you don't have to even know n-value to play it!
Edit:
Just to let you know: I created a list of 256-bit curves, with p-values, b-values, and n-values, as close to secp256k1, as I could.
I am recalculating them, because I forgot that p%4==3. But I will publish recalculated version soon.
