# Multiplication of a Pedersen commitment 10 times

Suppose I have a Pedersen Commitment $$cm(x,r) = g^xh^r$$ where $$g,h$$ are generators of group $$G$$ of prime order $$p$$. Based on $$cm$$, I want to create a commitment $$cm' = (10\cdot x,r')$$, without of course opening the original commitment, while knowing that $$x$$ is a relatively small value (e.g. less than 1 million) through a range proof.

Based on the additive homomorphic property, I was thinking of just multiplying $$cm$$ by itself 10 times, i.e. computing $$cm^{10}$$ which would open with $$(10\cdot x, 10\cdot r)$$. Is there anything that can go wrong with this?

• I think the question is missing context. What is your actual goal? Who creates G and the generators? And what are you using the commitments for? Just exponentiating an elemnt is something everyone can do. And it's fairly obvious, that the modified opening works. Btw: the assumption "x is small" indicates, that this is used in a wrong way. If x is chosen from a small subset, security is already gone.
– tylo
Dec 17 '20 at 1:03
• G and generators are public parameters. These commitments will be part of a zero-knowledge protocol proving that cm' hides a value 10 times larger than cm. By the assumption "x is small" I mean that it cannot be as large as p. e.g. you know that I hide a value between 0 and 1 million through a range proof. In my context there is no security issue with that. Dec 17 '20 at 1:15
• @tylo: "If x is chosen from a small subset, security is already gone" - that is incorrect; even if someone knows apriori that a commitment holds one of two values, they cannot tell which it is. Dec 17 '20 at 3:29
• @poncho You're right, my mistake. I somehow thought about the random value being from a small domain.
– tylo
Dec 17 '20 at 23:50