Secure computation of the k^th-Ranked element

Given two or more parties possessing large, confidential datasets, we consider the problem of securely computing the k^th-ranked element of the union of the datasets, e.g. the median of the values in the datasets. We investigate protocols with sublinear computation and communication costs. In the two-party case, we show that the k^th-ranked element can be computed in log k rounds, where the computation and communication costs of each round are O(log M), where log M is the number of bits needed to describe each element of the input data. The protocol can be made secure against a malicious adversary, and can hide the sizes of the original datasets. In the multi-party setting, we show that the k^th-ranked element can be computed in log M rounds, with O(s log M) overhead per round, where s is the number of parties. The multi-party protocol can be used in the two-party case and can also be made secure against a malicious adversary.