I am curious that how to deal with the message or randomness overflow in Pedersen commitment?
For more details:
For EC Pedersen commitment: The two generators are G and H.
Two messages and randomness are $m_1$, $m_2$, $r_1$, $r_2$, so the two Pedersen commitments are $Gm_1+Hr_1$ and $Gm_2+Hr_2$
If adding these two, I got a new Pedersen commitment as $G(m_1+m_2)+H(r_1+r_2)$ with message $m_1+m_2$ and randomness $r_1+r_2$.
But then what if the message $m_1+m_2$(or randomness $r_1+r_2$) overflows?
For example messages are in field mod 2^64, than if message becomes some 2^64+1, it would become 1.
As G*(2^64+1) should not equal to G*1, unless G has the order of 2^64.
So I am curious about how it works.
Any help would be truly appreciated.