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The usual Pedersen Commitment looks like such:
C = xG + aH
Where x and a are scalars.
However, we can also commit vectors, s.t.
C = xG + (a_1 * H_1 + a_2 * H_2)
Could anyone point out why we change the H values for each vector value (a_1 and a_2)? To my knowledge, having the same H for a_1 and a_2 would still be perfectly hiding.
a_1 means 'a subscript 1' denoting the first scalar value in the vector a.
I believe H_i is calculated s.t. H_i = Hash(G + i) * G
It is correct that using the same H each time would be perfectly hiding. However, the job of a Pedersen Commitment is not only to hide, but to commit.
If you used the same H each time, then all you would be committing to would be the sum of all components. You could later pretend that you had committed to a different number of components or to different component values as long as they still added up to the same amount.
