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From the output produced by a memoryless deletion channel with a uniformlyrandom input of known length $n$, one obtains a posterior distribution on thechannel input. The difference between the Shannon entropy of this distributionand that of the uniform prior measures the amount of information about thechannel input which is conveyed by the output of length $m$. We firstconjecture on the basis of experimental data that the entropy of the posterioris 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 involvingbinary (sub/super)-sequences and prove the minimal entropy conjecture forsingle and double deletions using clustering techniques. We then prove theminimization conjecture in the asymptotic limit using results from hidden wordstatistics by showing how the analytic-combinatorial methods of Flajolet,Szpankowski and Vall\'ee, relying on generating functions, can be applied toresolve the case of fixed output length and $n\rightarrow\infty$.  Next, we revisit the notion of deniability in quantum key exchange (QKE). Weintroduce and formalize the notion of coercer-deniable QKE. We then establish aconnection between covert communication and deniability to propose DC-QKE, asimple and provably secure construction for coercer-deniable QKE. We relatedeniability to fundamental concepts in quantum information theory and suggest ageneric approach based on entanglement distillation for achievinginformation-theoretic deniability, followed by an analysis of other closelyrelated results such as the relation between the impossibility ofunconditionally secure quantum bit commitment and deniability. Finally, wepresent an efficient coercion-resistant and quantum-secure voting scheme, basedon fully homomorphic encryption.

From Information Theory Puzzles in Deletion Channels to Deniability in Quantum Cryptography