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− | Cryptogenography | + | |
− | In this problem, there are $k$ players and an eavesdropper. Input is a random bit $X \in \{0,1\}$, also called | + | Cryptogenography is concerned with the following question. "How to share a secret without revealing the secret owner?" Please see Brody et al. {{cite|BrodyJSW-14}}. |
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+ | In this problem, there are $k$ players and an eavesdropper. Input is a random bit $X \in \{0,1\}$, also called "the secret." The secret owner $J$ is chosen uniformly at random from $[k]$. Players have private randomness and they can communicate publicly on a shared blackboard visible to everyone. Players are said to win if | ||
* everyone learns the secret, and | * everyone learns the secret, and | ||
* eavesdropper does not guess the secret owner correctly. | * eavesdropper does not guess the secret owner correctly. | ||
− | We are interested in maximizing the probability that players win (maximum success probability). For $k=2$, the following trivial protocol has success probability $0.25$. Just output $1$. It can be shown easily that maximum success probability is at most $0.5$. The bounds can be improved to $0.3384 \le $ maximum success probability $\le 0.361$. Here is a protocol that achieves success probability $1/3$. | + | We are interested in maximizing the probability that players win (maximum success probability). |
+ | |||
+ | For $k=2$, the following trivial protocol has success probability $0.25$. Just output $1$. It can be shown easily that maximum success probability is at most $0.5$. The bounds can be improved to $0.3384 \le $ maximum success probability $\le 0.361$. Here is a protocol that achieves success probability $1/3$. | ||
+ | |||
+ | '''First round:''' | ||
+ | If Alice has the secret, | ||
+ | |||
+ | * with probability $2/3$ she asks Bob to speak second | ||
+ | * with probability $1/3$ she speaks second. | ||
+ | |||
+ | Else (vice-versa) | ||
+ | * with probability $2/3$ she speaks second | ||
+ | * with probability $1/3$ she asks Bob to speak second. | ||
+ | |||
+ | '''Second round:''' | ||
+ | If the speaker has the secret, announce it, else announce a random bit. | ||
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+ | |||
+ | |||
+ | For large $k$, the following bounds can be shown: $0.5644 \le $ maximum success probability $\le 0.75$. | ||
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− | + | Can we improve these bounds? | |
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− | + | For more information, including state of the art bounds, see: {{cite|Jakobsen-14|DoerrK-16}} | |
+ | <references /> |