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.