Open Problems:79

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

Cryptogenography, introduced by Brody et al. [BrodyJSW-14], is concerned with the following question: “How to share a secret without revealing the secret owner?” 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 decides that Bob speaks in the second round
• with probability $1/3$, she speaks in the second round.

Else (if she does not have the secret)

• with probability $1/3$, she decides that Bob speaks in the second round
• with probability $2/3$, she speaks in the second round.

Second round:

If the speaker has the secret, announce it. Otherwise, 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? For more information, including the state of the art bounds, see the papers of Jakobsen [Jakobsen-14] and Doerr and Kunnemann [DoerrK-16].