We study communication with consensus over a broadcast channel—the receivers reliably decode the sender’s message when the sender is honest, and their decoder outputs agree even if the sender acts maliciously. We characterize the broadcast channels which permit this Byzantine consensus and determine their capacity. We show that communication with consensus is possible only when the broadcast channel has embedded in it a natural “common channel” whose output both receivers can unambiguously determine from their own channel outputs. Interestingly, in general, the consensus capacity may be larger than the point-to-point capacity of the common channel, i.e., while decoding, the receivers may make use of parts of their output signals on which they may not have consensus provided there are some parts (namely, the common channel output) on which they can agree.

We study the distributed function computation problem with$k$users of which at most$s$may be controlled by an adversary and characterize the set of functions of the sources the decoder can reconstruct robustly in the following sense if the users behave honestly, the function is recovered with high probability (w.h.p.); if they behave adversarially, w.h.p, either one of the adversarial users is identified or the function is recovered with vanishingly small distortion.

We study a secret sharing problem with three secrets where the secrets are allowed to be related to each other, i.e., only certain combinations of the three secrets are permitted. The dealer produces three shares such that every pair of shares reveals a unique secret and reveals nothing about the other two secrets, other than what can be inferred from the revealed secret. For the case of binary secrets, we exactly determine the minimum amount of randomness required by the dealer, for each possible set of permitted combinations. Our characterization is based on new lower and upper bounds.

We study communication with consensus over a broadcast channel - the receivers reliably decode the sender’s message when the sender is honest, and their decoder outputs agree even if the sender acts maliciously. We characterize the broadcast channels which permit this byzantine consensus and determine their capacity.

We study the fundamental problem of index coding under an additional privacy constraint that requires each receiver to learn nothing more about the collection of messages beyond its demanded messages from the server and what is available to it as side information. To enable such private communication, we allow the use of a collection of independent secret keys, each of which is shared amongst a subset of users and is known to the server. The goal is to study properties of the key access structures that make the problem feasible and then design encoding and decoding schemes efficient in the size of the server transmission as well as the sizes of the secret keys. We call this theprivate index codingproblem. We begin by characterizing the key access structures that make private index coding feasible. We also give conditions to check if a given linear scheme is a valid private index code. For up to three users, we characterize the rate region of feasible server transmission and key rates, and show that all feasible rates can be achieved using scalar linear coding and time sharing; we also show that scalar linear codes are sub-optimal for four receivers. The outer bounds used in the case of three users are extended to arbitrary number of users and seen as a generalized version of the well-known polymatroidal bounds for the standard non-private index coding. We also show that the presence of common randomness and private randomness does not change the rate region. Furthermore, we study the case where the server has the ability to multicast to any subset of users, and demonstrate how this flexibility can be used to provide privacy and characterize the minimum number of server multicasts required.

We study private two-terminal hypothesis testing with simple hypotheses where the privacy goal is to ensure that participating in the testing protocol reveals little additional information about the other user's observation when a user is told what the correct hypothesis is. We show that, in general, meaningful correctness and privacy cannot be achieved if the users do not have access to correlated (but, not common) randomness. We characterize the optimal correctness and privacy error exponents when the users have access to non-trivial correlated randomness (those that permit secure multiparty computation).

We study the problem of index coding under the privacy requirement that receivers do not learn anything more than the messages they already have as side information and the message they want from the server. To achieve this private index coding, we consider the use of secret keys that are shared among various subsets of users and the server. We characterize key access structures that allow private index coding. For up to three receivers, we characterize the rate region of transmission and key rates and show that scalar coding is optimal; we also show that scalar linear codes are sub-optimal for four receivers. Furthermore, when no keys are available, we consider a weaker notion of privacy analogous to weak security. Finally, for a different setting in which the server is allowed to send messages exclusively to a subset of users, we study the number of transmissions required to achieve error-free decoding and privacy.