Problem 79: Cryptogenography
| 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 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? For more information, including state of the art bounds, see: [Jakobsen-14,DoerrK-16]
