I'm new to blind signatures, trying to understand security notions of it: blindness and one-more unforgeability.

I found out that in the security game for blindness property, the adversary $\mathcal{A}$ interacts with two users (each with messages $m_0$ and $m_1$). Without knowing who signed first, $\mathcal{A}$ gets a list of signatures in order $\{\sigma(m_b),\sigma(m_{1-b})\}$. Then $\mathcal{A}$ would try to guess which of them came first. That is, to guess a bit $b'$. So $\mathcal{A}$ succeeds if $b'=b$.

My question is: why do we need to interact with two users (or twice)? Can't we just give $\mathcal{A}$ a signature of randomly selected message ($m_0$ or $m_1$)? In that case, $\mathcal{A}$ would try to guess which of those two messages were signed.

Can anyone give me the reason why do we need two interactions in the security game?

Thanks in advance.


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