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ABE and OKAMOTO definition of partially blind signature scheme

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I'm trying to understand the definition of a partially blind scheme that is described with Game A presented in

Abe and Okamoto paper

Game A part 1

Game A prt 2

  1. In line 5 is $msg_0$ correct or should it be replaced by $(info_0,msg_b,sig_b)$? similarly for $U_1$. I'm confused since if $msg_0$ is always on the private tape of $U_0$ then S can always guess b correctly by checking if $sig_b$ verifies $msg_0$.

I'm missing the point here I'm certain. Thank you for helping out.

I sit in a Tesla and translated this thread with Ai:

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