Secure computation of randomized functions: Further results

We consider secure computation of randomized functions by two users, where both the users (Alice and Bob) have inputs, Alice sends a message to Bob over a rate-limited, noise-free link, and then Bob produces the output. We study this problem when privacy is required only against Bob, i.e., from the message, Bob must not learn any information about Alice's input other than what can be inferred by his own input and output. We give a single-letter expression for the optimal rate. We also explicitly characterize securely computable randomized functions when input has full support, which leads to a much simpler expression for the optimal rate. Recently, Data (ISIT 2016) studied the other two cases (first, when privacy is required against both the users; and second, when privacy is required only against Alice) and obtained single-letter expressions for optimal rates in both the scenarios. Yassaee, Gohari, and Aref (IEEE Transactions on Information Theory 2015) studied the case when there is no privacy requirement and obtained a single-letter expression for the optimal rate, when Alice and Bob interact for arbitrary but finite number of rounds, and both of them may produce potentially different outputs.