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arXiv:2105.01400

Coin Flipping of \emph{Any} Constant Bias Implies One-Way Functions

Itay Berman, Iftach Haitner, Aris Tentes

Published May 4, 2021• 0 citations
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Abstract

We show that the existence of a coin-flipping protocol safe against \emph{any} non-trivial constant bias (\eg $.499$) implies the existence of one-way functions. This improves upon a recent result of Haitner and Omri [FOCS '11], who proved this implication for protocols with bias $\frac{\sqrt2 -1}2 - o(1) \approx .207$. Unlike the result of Haitner and Omri, our result also holds for \emph{weak} coin-flipping protocols.

DOI
10.48550/arxiv.2105.01400

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Coin flipping of any constant bias implies one-way functions

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We show that the existence of a coin-flipping protocol safe against \emph{any} non-trivial constant bias (\eg .499.499.499) implies the existence of one-way functions. This improves upon a recent result of Haitner and Omri [FOCS '11], who proved this implication for protocols with bias 2−12−o(1)≈.207\frac{\sqrt2 -1}2 - o(1) \approx .20722​−1​−o(1)≈.207. Unlike the result of Haitner and Omri, our result also holds for \emph{weak} coin-flipping protocols.
It is well known (cf. Impagliazzo and Luby [in 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, 2−12−o(1)\frac{\sqrt2 -1}2 - o(1)22​−1​−o(1)) imply one-way functions.
Modern Cryptography is based on computational intractability assumptions, e.g., Factoring, Discrete Logarithm, Diffie-Helman etc. However, since an assumption might be proven incorrect, there has been a lot of focus in order to construct cryptographic primitives based on the possibly most minimal assumption. The most popular minimal assumption, which is implied by the existence of almost all cryptographic primitives, is the existence of One Way Functions. Coin-Flipping protocols are known to be implied by One-Way Functions, however, a complete characterization of the inverse direction is not known. There was even speculation that weak notions of Coin Flipping Protocols might be strictly weaker than One Way Functions. In this thesis we show that even very weak notions of Coin Flipping protocols do imply One Way Functions. In particular we show that the existence of a coin-flipping protocol safe against any non-trivial constant bias (e.g., .499) implies the existence of One Way Functions. This improves upon a recent result of Haitner and Omri [FOCS ’11], who proved this implication for protocols with bias √ 2−1 2 − o(1) ≈ .207. Unlike the former result result, our result also holds for weak coin-flipping protocols.
It is well known (cf., Impagliazzo and Luby [FOCS '89]) 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 [FOCS '10] proved the same for constant-round protocols (with any non-trivial 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, (√2 -1)/2 - o(1)) imply one-way functions.