How to Bootstrap Anonymous Communication

We ask whether it is possible to anonymously communicate a large amount of data using only public (non-anonymous) communication together with a small anonymous channel. We think this is a central question in the theory of anonymous communication and to the best of our knowledge this is the first formal study in this direction. To solve this problem, we introduce the concept of anonymous steganography: think of a leaker Lea who wants to leak a large document to Joe the journalist. Using anonymous steganography Lea can embed this document in innocent looking communication on some popular website (such as cat videos on YouTube or funny memes on 9GAG). Then Lea provides Joe with a short key $k$ which, when applied to the entire website, recovers the document while hiding the identity of Lea among the large number of users of the website. Our contributions include: - Introducing and formally defining anonymous steganography, - A construction showing that anonymous steganography is possible (which uses recent results in circuits obfuscation), - A lower bound on the number of bits which are needed to bootstrap anonymous communication.

Timeability of Extensive-Form Games

Extensive-form games constitute the standard representation scheme for games with a temporal component. But do all extensive-form games correspond to protocols that we can implement in the real world? We often rule out games with imperfect recall, which prescribe that an agent forget something that she knew before. In this paper, we show that even some games with perfect recall can be problematic to implement. Specifically, we show that if the agents have a sense of time passing (say, access to a clock), then some extensive-form games can no longer be implemented; no matter how we attempt to time the game, some information will leak to the agents that they are not supposed to have. We say such a game is not exactly timeable. We provide easy-to-check necessary and sufficient conditions for a game to be exactly timeable. Most of the technical depth of the paper concerns how to approximately time games, which we show can always be done, though it may require large amounts of time. Specifically, we show that for some games the time required to approximately implement the game grows as a power tower of height proportional to the number of players and with a parameter that measures the precision of the approximation at the top of the power tower. In practice, that makes the games untimeable. Besides the conceptual contribution to game theory, we believe our methodology can have applications to preventing information leakage in security protocols.

A numbers-on-foreheads game

Is there a joint distribution of $n$ random variables over the natural numbers, such that they always form an increasing sequence and whenever you take two subsets of the set of random variables of the same cardinality, their distribution is almost the same? We show that the answer is yes, but that the random variables will have to take values as large as $2^{2^{dots ^{2^{Thetaleft(frac{1}{epsilon}right)}}}}$, where $epsilonleq epsilon_n$ measures how different the two distributions can be, the tower contains $n-2$ $2$s and the constants in the $Theta$ notation are allowed to depend on $n$. This result has an important consequence in game theory: It shows that even though you can define extensive form games that cannot be implemented on players who can tell the time, you can have implementations that approximate the game arbitrarily well.