In Bulletproofs range proof, Pedersen commitments look like $C=aG+bH$
If I need to proof several values' range proof and keep G as constant, do I need to change H everytime?
I have found it doesn't change in Grin and dalek-cryptography.
Pedersen commitments fulfil two basic properties:
Since $b$ is uniformly random, $C=aG+bH$ can become any element of the group, independent from the randomness of $a$.
If I don't know the discrete logarithm relation between $G$ and $H$, I cannot cheat by opening $C$ to a different $a,b$ pair.
Additionally, Bulletproofs relies on Pedersen commitments being additively homomorphic: for another commitment $C'=a'G+b'H$, we have that $C+C'$ is a commitment to $a+a',b+b'$. To see this: $C+C' = (a+a')G+(b+b')H$. For this to work, we need $H$ to be a fixed point.
So, to sum up: $H$ shouldn't change because we require additive homomorphism, and $H$ does not need to change because it already provides perfect hiding!