Difference between revisions of "Open Problems:79"

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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
 
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.
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We are interested in maximizing the probability that players win (maximum success probability).
 
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 $\le 0.361$.
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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)
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* with probability $2/3$ she speaks second
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* with probability $1/3$ she asks Bob to speak second.
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'''Second round:'''
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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 $\le 0.75$.
 
For large $k$, the following bounds can be shown:  $0.5644 \le $ maximum success probability $\le 0.75$.
 +
  
 
Can we improve these bounds?
 
Can we improve these bounds?

Revision as of 21:45, 31 March 2017

Suggested by Joshua Brody
Source Banff 2017
Short link https://sublinear.info/79

Cryptogenography is concerned with the following question. "How to share a secret without revealing the secret owner?" Please see Brody et al. [BJSW-14] and Jakobsen [J-16].

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
  • 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 $1/3$.

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.


For large $k$, the following bounds can be shown: $0.5644 \le $ maximum success probability $\le 0.75$.


Can we improve these bounds?