来自宇宙源的硬币翻转：关于真正随机比特的error correction
Coin flipping from a cosmic source: On error correction of truly random bits
摘要：
我们研究了一个与抛硬币、编码理论和噪声敏感性有关的问题。考虑真正随机比特$x \in \bits^n$的源和$k$方，它们具有源比特$y^i \in \bits^n$的噪声版本，其中对于所有$i$和$j$，它保持$\Pr[y^i_j = x_j] = 1 - \eps$独立于所有$i$和$j$。也就是说，每一方以概率$1-ε$正确地看到每一位，并且以概率$ε$不正确地（翻转）看到每一位，对于所有位和所有方都是独立的。不能通信的各方希望事先就{ \em平衡}函数$f_i : \bits^n \to \bits$达成一致，使得$\Pr[f_1(y^1) = ... = f_k(y^k)]$最大化。换句话说，每一方都想抛一枚公平的硬币，这样所有各方都有相同硬币的概率最大化。函数$f_i$可以被认为是源$x$的纠错过程。当$k=2,3$没有纠错是可能的，因为最佳协议由$f_i(x^i) = y^i_1$给出。另一方面，对于大的$k$值，存在更好的协议。我们研究了最优协议的一般性质和问题关于$k$、$n$和$\eps$的渐近行为。我们的分析使用的工具，从概率，离散傅立叶分析，凸性和离散对称化。
英文摘要：
We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits $x \in \bits^n$, and $k$ parties, who have noisy versions of the source bits $y^i \in \bits^n$, where for all $i$ and $j$, it holds that $\Pr[y^i_j = x_j] = 1 - \eps$, independently for all $i$ and $j$. That is, each party sees each bit correctly with probability $1-ε$, and incorrectly (flipped) with probability $ε$, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on {\em balanced} functions $f_i : \bits^n \to \bits$ such that $\Pr[f_1(y^1) = ... = f_k(y^k)]$ is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The functions $f_i$ may be thought of as an error correcting procedure for the source $x$. When $k=2,3$ no error correction is possible, as the optimal protocol is given by $f_i(x^i) = y^i_1$. On the other hand, for large values of $k$, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to $k$, $n$ and $\eps$. Our analysis uses tools from probability, discrete Fourier analysis, convexity and discrete symmetrization.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
