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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.

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  • $\begingroup$ The points $G$ and $H$ are usually public parameters so that the discrete logarithm from $G$ to $H$ is not known. This leads to them being the same for every commitment (at least usually). $\endgroup$ – VincBreaker May 5 at 15:18
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Pedersen commitments fulfil two basic properties:

  1. Perfectly hiding.

Since $b$ is uniformly random, $C=aG+bH$ can become any element of the group, independent from the randomness of $a$.

  1. Computationally binding.

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!

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