How to get the generator G point for any new curve?

Question

I have this bitcoin curve:

y^2 = x^3 + 7

but the finite field Fp is modified to be:

n=115792089237316195423570985008687907853269984665640564039457584007908834675927

1-What is the G point?

2-how to get the Y coordinate for any X value?

I'm using this C# code to get Y from X but it works with original bitcoin curve only.

    public BigInteger mod(BigInteger num, BigInteger by) {
        BigInteger res = num % by;
        if (res < 0) { res += by; }
        return res;
    }

    public BigInteger EC_GetY(BigInteger x, bool modIt, BigInteger n) {
        BigInteger n_OVER_FOUR = (n + 1) / 4;
        BigInteger alpha = mod(BigInteger.Pow(x, 3) + 7, n);
        BigInteger beta = BigInteger.ModPow(alpha, n_OVER_FOUR, n);//SqRtN
        BigInteger Y = beta;

        if (!modIt) { return Y; }
        Y = n - beta;
        return Y;
    }

Answer 1

You can't just change N. N is the number of elements in the finite group. You can pick a new (prime) P close to 2^256 (or however many bits you want) and get a resulting N.

1) G is a point in the group that when added to itself produces another point in the group. Hence it can generate all other points with repeated additions (generator). If the number of elements in the group is prime then every point has this property and can be used as a generator (via Lagrange's Theorem).

2) Just use the curve equation. y^2 = x^3 + 7 => y = modsqrt(x^3 + 7) (everything is mod P)

Answer 2

G is just an arbitrarily selected point from the largest sub-group (in the case of secp256k1, there is only one sub-group).

It is slightly preferable if G is selected in a rigid way that shows you it wasn't constructed so that you'd know the discrete log with respect to some other point.

E.g. x=1 is on your curve, so you could use {1, 2455109838632409200075678375825088630665501513214133143088640413693816705352929} as your generator.