For example, I could use:
If the discrete log is already backdoored with the standard base point $G$, then changing the base to another point on the curve doesn't solve this issue.
Let you know that $G$ is backdoored and you changed the base to $G' \neq G$. Then the entity that created the backdoor can use this to find the private keys.
Let $P = [k]G'$ be a public key with the new base. The attacker solves Dlog of $G' = [a]G$ only once. Using this they forms $P = [ak]G$. This is in the backdoored base so that they can solve the discrete logairhtmm to find $ak$. Once $ak$ is found, extracting the secret key can be performed with a simple modular arithmetic $k = ak \cdot a^{-1} \bmod n$ where the $a^{-1}$ is the inverse of $a$ in the modulo $n$.
As a result, once you have a backdoored discrete logarithm, then the curve is not safe to use. It is all in one, if a base point has a trapdoor then all base points have trapdoors!
However, if I change any of those parameters and used them, then will the security of the trapdoor function be compromised significantly?
Changing the parameters $p,a$, and $b$ that defines $n$ and $h$, except the basepoint, change the curve and the new curve needs to be extensively analyzed;
- Does the curve order has a prime or has a large prime factor?
- Does the twist of the curve have large prime order?
- Does it have a safe discrete log?
- ...
These are the basics, more on this see safecurves
