How to multiply the Pedersen Commitment of two numbers?

Given two numbers $$x_1$$, $$x_2$$ and their respective binding numbers, $$b_1$$ and $$b_2$$, let's take their Pedersen Commitment to be $$C(x_n, b_n)$$ $$\forall n=1,2$$.

What is $$C(x_1 * x_2, b_1 * b_2)$$?

Sort of related, we have the identity $$C(x_1*x_2,-b_1*b_2)=x_1C(x_2,b_2)-b_2C(x_1,b_1)=x_2C(x_1,b_1)-b_1C(x_2,b_2)$$ (note the similarity to the Diffie-Hellman identity) but in general we cannot compute a Pedersen commitment of the product of two numbers from their Pedersen commitments without knowledge of the hidden values.

• Thanks for sharing!
– Jim
Mar 9 at 15:50