Given A, B, G, H prove the I know x s.t. $A=xG$ and $B=xH$

Question

Given A, B, G, H .How would I prove that I know x such that $A=xG$ and $B=xH$

Answer 1

There is a standard $\Sigma$-protocol for proving knowledge of a witness showing that $(G,H,A,B)$ is a DDH-tuple, assuming a group of prime order $p$:

• The prover picks a random exponent $r \in \mathbb{Z}_p$ and sends $(R,S) = (rG, rH)$.
• The verifier sends a random challenge $e$ from $\mathbb{Z}_p$.
• The prover computes and send $d = e\cdot x + r$.
• The verifier accepts the proof if and only if $eA+R = dG$ and $eB+S=dH$.

Since you seem to ask these questions for the purpose of learning, I'd suggest that you try to prove yourself the security of this protocol (that is, soundness, and honest-verifier zero-knowledge). You can take inspiration from the security analysis I gave for a similar protocol in this answer, that deals with the Schnorr protocol (about which you asked in another question).