# Time-Lock Puzzles in the Random Oracle Model

Mohammad Mahmoody, Tal Moran, Salil Vadhan

### Abstract

A time-lock puzzle is a mechanism for sending messages “to the future”. The sender publishes a puzzle whose solution is the message to be sent, thus hiding it until enough time has elapsed for the puzzle to be solved. For time-lock puzzles to be useful, generating a puzzle should take less time than solving it. Since adversaries may have access to many more computers than honest solvers, massively parallel solvers should not be able to produce a solution much faster than serial ones.

To date, we know of only one mechanism that is believed to satisfy these properties: the one proposed by Rivest, Shamir and Wagner (1996), who originally introduced the notion of time-lock puzzles. Their puzzle is based on the serial nature of exponentiation and the hardness of factoring, and is therefore vulnerable to advances in factoring techniques (as well as to quantum attacks). In this work, we study the possibility of constructing time-lock puzzles in the random-oracle model. Our main result is negative, ruling out time-lock puzzles that require more parallel time to solve than the total work required to generate a puzzle. In particular, this should rule out black-box constructions of such timelock puzzles from one-way permutations and collision-resistant hash-functions. On the positive side, we construct a time-lock puzzle with a linear gap in parallel time: a new puzzle can be generated with one round of n parallel queries to the random oracle, but n rounds of serial queries are required to solve it (even for massively parallel adversaries).

### Versions

- To appear in
*Advances in Cryptology --- CRYPTO `11*, volume 6841 of Lecture Notes in Computer Science, pages 39-50. Springer-Verlag, 14-18 August 2011. [pdf]

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