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Public parameters (or system parameters as you call them) can be distributed however you want. Papers which introduce Identity-based Encryption schemes usually only present the scheme and not how it can be applied in the real world 1. The public parameters in IBE can be compared to the pre-shared data in the Diffie-Hellman key exchange protocol where both ...

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(I assume you are asking about this paper based on your previous questions.) They do not prove BDDH in their security proofs. Assume that an attacker $A$ breaks IND-sMID-CPA of the above scheme with probability greater than $\epsilon$ within time $t$ [...]. We show that using $A$, one can construct an attacker $B$ for solving the BDDH problem. In ...

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Sometimes, to make it easier to understand, the advantage is multiplied by a factor of 2: $Adv = 2|\Pr[b' = b] - 1/2|$. Note that when the adversary has no advantage in breaking the protocol, the only thing s/he can do is guessing $b'$ at random. In this case, $\Pr[b' = b] = 1/2$ and thus $Adv = 0$. When the adversary always correctly finds the value ...

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That property lets a trusted n2-byte random string be enough to make the rest of public keys fit into n bytes. In particular, forward secrecy can be more efficient if the sender can store such a string, since the string can have been generated by the same party as generates the rest of the public keys. Also, if for random private keys and independent n2-...

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