From the output produced by a memoryless deletion channel with a uniformly random input of known length $n$, one obtains a posterior distribution on the channel input. The difference between the Shannon entropy of this distribution and that of the uniform prior measures the amount of information about the channel input which is conveyed by the output of length $m$. We first conjecture on the basis of experimental data that the entropy of the posterior is minimized by the constant strings $\texttt{000}\ldots$, $\texttt{111}\ldots$ and maximized by the alternating strings $\texttt{0101}\ldots$, $\texttt{1010}\ldots$. We present related combinatorial theorems involving binary (sub/super)-sequences and prove the minimal entropy conjecture for single and double deletions using clustering techniques. We then prove the minimization conjecture in the asymptotic limit using results from hidden word statistics by showing how the analytic-combinatorial methods of Flajolet, Szpankowski and Vall\'ee, relying on generating functions, can be applied to resolve the case of fixed output length and $n\rightarrow\infty$. Next, we revisit the notion of deniability in quantum key exchange (QKE). We introduce and formalize the notion of coercer-deniable QKE. We then establish a connection between covert communication and deniability to propose DC-QKE, a simple and provably secure construction for coercer-deniable QKE. We relate deniability to fundamental concepts in quantum information theory and suggest a generic approach based on entanglement distillation for achieving information-theoretic deniability, followed by an analysis of other closely related results such as the relation between the impossibility of unconditionally secure quantum bit commitment and deniability. Finally, we present an efficient coercion-resistant and quantum-secure voting scheme, based on fully homomorphic encryption.

中文翻译：

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从删除通道中的信息论谜题到量子密码学中的可否认性

从具有已知长度 $n$ 的均匀随机输入的无记忆删除通道产生的输出中，可以获得通道输入的后验分布。该分布的香农熵与均匀先验的香农熵之间的差异衡量了有关通道输入的信息量，该信息量由长度为 $m$ 的输出所传达。我们首先根据实验数据推测后验的熵通过常量字符串 $\texttt{000}\ldots$, $\texttt{111}\ldots$ 最小化，并通过交替字符串 $\texttt{ 最大化0101}\ldots$, $\texttt{1010}\ldots$. 我们提出了涉及二元（子/超）序列的相关组合定理，并使用聚类技术证明了单删除和双删除的最小熵猜想。然后，我们通过展示 Flajolet、Szpankowski 和 Vall\'ee 的分析组合方法如何应用依赖生成函数来解决固定输出长度的情况，使用隐藏词统计的结果证明渐近极限中的最小化猜想和 $n\rightarrow\infty$。接下来，我们重新审视量子密钥交换 (QKE) 中的可否认性概念。我们引入并形式化了 coercer-deniable QKE 的概念。然后，我们在隐蔽通信和可否认性之间建立联系，以提出 DC-QKE，这是一种用于强制可否认 QKE 的简单且可证明安全的结构。我们将可否认性与量子信息理论中的基本概念联系起来，并提出了一种基于纠缠蒸馏的通用方法，以实现信息理论的可否认性，随后分析了其他密切相关的结果，例如无条件安全量子比特承诺的不可能性与可否认性之间的关系。最后，我们提出了一种基于全同态加密的高效抗强制和量子安全投票方案。