# Coin Flipping with Constant Bias Implies One-Way Functions

## Abstract

*Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science*, 1989, pp. 230--235]) that the existence of almost all “interesting" cryptographic applications, i.e., ones that cannot hold information theoretically, implies one-way functions. An important exception where the above implication is not known, however, is the case of coin-flipping protocols. Such protocols allow honest parties to mutually flip an unbiased coin, while guaranteeing that even a cheating (efficient) party cannot bias the output of the protocol by much. Impagliazzo and Luby proved that coin-flipping protocols that are safe against negligible bias do imply one-way functions, and, very recently, Maji, Prabhakaran, and Sahai [in

*Proceedings of the 2001 51st Annual IEEE Symposium on Foundations of Computer Science*, 2010, pp. 613--622] proved the same for constant-round protocols (with any nontrivial bias). For the general case, however, no such implication was known. We make progress towards answering the above fundamental question, showing that (strong) coin-flipping protocols safe against a constant bias (concretely, $\frac{\sqrt2 -1}2 - o(1)$) imply one-way functions.

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**Submitted**: 24 August 2012

**Accepted**: 6 November 2013

**Published online**: 4 March 2014

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