Is it secure to use ECDSA for any arbitrary point on the Elliptic Curve as the Generator point? My question concerns the elliptic curve $E$ over a prime field $\mathbb F_p$. To the best of my understanding, ECDSA requires a Generator point $G$ of prime order $n$, and the $r$ and $s$ values of the signature must be within the range $[1,n-1]$. The concept of ECDSA is to show a person that one is in possession of a secret $k$ s.t. $kG = K$, without (obviously) revealing $k$.

However, suppose that you are aware of public key $K$ and are looking to do a tweak of $K$. Let $t$ be the tweaking factor, and so $tK = K'$. If you then wanted to prove that you are in possession of $t$, how would you do that?

An immediate thought (which I am confident will not work) is to use ECDSA with $K$ as the generator point. However, it might be the case that $K$ is of a very small order $n$ which may not be prime. 

If someone can shed light on how to prove that one is in possession of secret $t$, I would be very happy. Also, the verifying party is in possession of both $k, K$, the secret key and public key pair.
However, suppose that you are aware of public key $K$ and are looking to do a tweak of $K$. Let $t$ be the tweaking factor, and so $tK=K'$. If you then wanted to prove that you are in possession of $t$, how would you do that?


The obvious way to do this is a Schnorr proof of knowledge, which does precisely what you're looking for; given a public $K, tK$, it demonstrates that you know $t$.  Wikipedia lists the interactive version; it can be made noninteractive by having the prover generate $c$ based on a hash of the commitment $t$ (and possibly other stuff if you need to bind the proof to something).


  An immediate thought (which I am confident will not work) is to use ECDSA with $K$ as the generator point. However, it might be the case that $K$ is of a very small order $n$ which may not be prime. 


That would also work (which isn't that surprising; ECDSA is essentially based on Schnorr, tweaked enough to avoid patents that have since expired); it's kludgier, however it might be preferable if you just happen to have an ECDSA implementation just lying around.  If $K$ does have small factors, you might need to work around it (as some values might not be invertable), however that wouldn't appear to be a major issue.

And, if $K$ is a very small order, then deriving $t$ would be easy, no matter what proof-of-knowledge method you use (and so there's little point in the problem).  In any case, we almost always use elliptic curves where the order of the curve is $hq$, where $h$ is a small value (the most common values are 1 and 8), and $q$ is a large prime - except for a handful of points (actually, a total of $h$ points), all points have a large order (at least $q$).