TCC 2014: Theory of Cryptography pp 217-239

# Can Optimally-Fair Coin Tossing Be Based on One-Way Functions?

• Dana Dachman-Soled
• Tal Malkin
Conference paper

DOI: 10.1007/978-3-642-54242-8_10

Part of the Lecture Notes in Computer Science book series (LNCS, volume 8349)
Cite this paper as:
Dachman-Soled D., Mahmoody M., Malkin T. (2014) Can Optimally-Fair Coin Tossing Be Based on One-Way Functions?. In: Lindell Y. (eds) Theory of Cryptography. TCC 2014. Lecture Notes in Computer Science, vol 8349. Springer, Berlin, Heidelberg

## Abstract

Coin tossing is a basic cryptographic task that allows two distrustful parties to obtain an unbiased random bit in a way that neither party can bias the output by deviating from the protocol or halting the execution. Cleve [STOC’86] showed that in any r round coin tossing protocol one of the parties can bias the output by Ω(1/r) through a “fail-stop” attack; namely, they simply execute the protocol honestly and halt at some chosen point. In addition, relying on an earlier work of Blum [COMPCON’82], Cleve presented an r-round protocol based on one-way functions that was resilient to bias at most $$O(1/\sqrt r)$$. Cleve’s work left open whether ”‘optimally-fair’” coin tossing (i.e. achieving bias O(1/r) in r rounds) is possible. Recently Moran, Naor, and Segev [TCC’09] showed how to construct optimally-fair coin tossing based on oblivious transfer, however, it was left open to find the minimal assumptions necessary for optimally-fair coin tossing. The work of Dachman-Soled et al. [TCC’11] took a step toward answering this question by showing that any black-box construction of optimally-fair coin tossing based on a one-way functions with n-bit input and output needs Ω(n/logn) rounds.

In this work we take another step towards understanding the complexity of optimally-fair coin-tossing by showing that this task (with an arbitrary number of rounds) cannot be based on one-way functions in a black-box way, as long as the protocol is ”‘oblivious’” to the implementation of the one-way function. Namely, we consider a natural class of black-box constructions based on one-way functions, called function oblivious, in which the output of the protocol does not depend on the specific implementation of the one-way function and only depends on the randomness of the parties. Other than being a natural notion on its own, the known coin tossing protocols of Blum and Cleve (both based on one-way functions) are indeed function oblivious. Thus, we believe our lower bound for function-oblivious constructions is a meaningful step towards resolving the fundamental open question of the complexity of optimally-fair coin tossing.

### Keywords

Coin-Tossing One-Way Functions Black-Box Separations

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© International Association for Cryptologic Research 2014

## Authors and Affiliations

• Dana Dachman-Soled
• 1