Assume there is a data stream of elements and a window of size m. Sliding window algorithms compute various statistic functions over the last m elements of the data stream seen so far. The time complexity of a sliding window algorithm is measured as the time required to output an updated statistic function value every time a new element is read. For example, it is well known that computing the sliding window maximum/minimum has time complexity O(1) while computing the sliding window median has time complexity O(log m). In this paper we close the gap between these two cases by (1) presenting an algorithm for computing the sliding window k-th smallest element in O(log k) time and (2) prove that this time complexity is optimal.

In the conditional disclosure of secrets (CDS) problem [Gertner et al., J. Comput. System Sci., 60 (2000), pp. 592--629] Alice and Bob, who hold inputs $x$ and $y$, respectively, wish to release a common secret $s$ to Carol (who knows both $x$ and $y$) if and only if the input $(x,y)$ satisfies some predefined predicate $f$. Alice and Bob are allowed to send a single message to Carol which may depend on their inputs and some joint randomness and the goal is to minimize the communication complexity while providing information-theoretic security. In this work, we initiate the study of CDS manipulation techniques and derive the following positive and negative results: (Closure) A CDS for $f$ can be turned into a CDS for its complement $\bar{f}$ with only a minor blow-up in complexity. More generally, for a (possibly nonmonotone) predicate $h$, we obtain a CDS for $h(f_1,\ldots,f_m)$ whose cost is essentially linear in the formula size of $h$ and polynomial in the CDS complexity of $f_i$. (Amplification) It is possible to reduce the privacy and correctness error of a CDS from constant to $2^{-k}$ with a multiplicative overhead of $O(k)$. Moreover, this overhead can be amortized over $k$-bit secrets. (Amortization) Every predicate $f$ over $n$-bit inputs admits a CDS for multibit secrets whose amortized communication complexity per secret bit grows linearly with the input length $n$ for sufficiently long secrets. In contrast, the best known upper-bound for single-bit secrets is exponential in $n$. (Lower-bounds) There exists a (nonexplicit) predicate $f$ over $n$-bit inputs for which any perfect (single-bit) CDS requires communication of at least $\Omega(n)$. This is an exponential improvement over the previously known $\Omega(\log n)$ lower-bound. (Separations) There exists an (explicit) predicate whose CDS complexity is exponentially smaller than its randomized communication complexity. This matches a lower-bound of Gay, Kerenidis, and Wee [ Advances in Cryptology, Lecture Notes in Comput. Sci. 9216, Springer, New York, 2015, pp. 485--502] and, combined with another result of theirs, yields an exponential separation between the communication complexity of linear CDS and non-linear CDS. This is the first provable gap between the communication complexity of linear CDS (which captures most known protocols) and nonlinear CDS.

In this paper we focus on sender-anonymous channels (a.k.a. Dining Cryptographers networks) and present a construction requiring a very low (constant) number of rounds of interaction while tolerating actively malicious behavior by some of the participants (up to less than half of them). Our construction is unconditionally secure (meaning that no bounds are placed on the computational power of the adversary), makes black-box use of a verifiable secret sharing (VSS) protocol, and is based on a special-purpose secure multiparty computation protocol implementing the method of "throwing darts;" its round complexity is essentially equal to that of the VSS protocol.