I would like to know the relation between p and n, because it seems that G is irrelevant in the curve calculations.
I can even give you some code to get n-value, based on p-value, and the curve equation. Of course, it is the simplest brute-force, and it will stop working if you use it on bigger numbers, but it is simple enough to understand it, and implement in any programming language you want.
#include <iostream>
int get_n_from_p(int p)
{
int n=1; //we start from n=1, and not n=0, because of the point at infinity
for(int y=0;y<p;++y)
{
for(int x=0;x<p;++x)
{
int y_square=(y*y)%p;
int x_cube=(x*x*x)%p;
int x_cube_plus_seven=(x_cube+7)%p;
if(y_square==x_cube_plus_seven)
{
++n;
}
}
}
return n;
}
int main()
{
for(int p=1;p<=1000;++p)
{
std::cout<<"p="<<p<<", n="<<get_n_from_p(p)<<'\n';
}
}
See? No generator is used there. Every single point is checked, and you can just create an image, and put a black pixel, if there is no point on a given curve, or put a white pixel, when you can find a match. This is basically what vjudeu did in his repository. As you can see, in this brute-force algorithm, there is no need to even pick any specific point, because all of them are checked.
ok? the question is how in 199x years it was possible to do it(brute-force). how this n and p was chosen? thanks