Robust Coin Flipping

TL;DR

This paper characterizes the possible biases in secure remote coin flipping protocols with multiple sources, showing how privacy constraints limit the achievable bias based on the number of sources and colluding parties.

Contribution

It provides a complete characterization of the biases achievable in secure coin flipping with multiple sources under different privacy constraints, improving upon prior work by Yao.

Findings

Bias can be any rational number when half or more sources are compromised.

Bias can be any algebraic number when fewer than half sources are compromised.

The results have applications in secure multiparty computation.

Abstract

Alice seeks an information-theoretically secure source of private random data. Unfortunately, she lacks a personal source and must use remote sources controlled by other parties. Alice wants to simulate a coin flip of specified bias $\alpha$, as a function of data she receives from $p$ sources; she seeks privacy from any coalition of $r$ of them. We show: If $p/2 \leq r < p$, the bias can be any rational number and nothing else; if $0 < r < p/2$, the bias can be any algebraic number and nothing else. The proof uses projective varieties, convex geometry, and the probabilistic method. Our results improve on those laid out by Yao, who asserts one direction of the $r=1$ case in his seminal paper [Yao82]. We also provide an application to secure multiparty computation.