The prophet inequalities problem has received significant study over the past decades and has several applications such as to online auctions. In this paper, we study two variants of the i.i.d. prophet inequalities problem, namely the windowed prophet inequalities problem and the batched prophet inequalities problem. For the windowed prophet inequalities problem, we show that for window size o(n), the optimal competitive ratio is α ≈ 0.745, the same as in the non-windowed case. In the case where the window size is n/k for some constant k, we show that α k < W IN n/k ≤ α k +o k (1) where W IN n/k is the optimal competitive ratio for the window size n/k prophet inequalities problem and α k is the optimal competitive ratio for the k sample i.i.d. prophet inequalities problem. Finally, we prove an equivalence between the batched prophet inequalities problem and the i.i.d. prophet inequalities problem.
Conditional disclosure of secrets (CDS) allows multiple parties to reveal a secret to a third party if and only if some pre-decided condition is satisfied. In this work, we bolster the privacy guarantees of CDS by introducing function-private CDS wherein the pre-decided condition is never revealed to the third party. We also derive a function secret sharing scheme from our function-private CDS solution. The second problem that we consider concerns threshold distributed point functions, which allow one to split a point function such that at least a threshold number of shares are required to evaluate it at any given input. We consider a setting wherein a point function is split among a set of parties such that multiple evaluations do not leak non-negligible information about it. Finally, we present a provably optimal procedure to perform threshold function secret sharing of any polynomial in a finite field. † The is the full version of the paper that will appear in Cryptology and Network Security (CANS), 2021.
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