Secret Communication over Broadcast Erasure Channels with State-feedback

We consider a 1-to-$K$ communication scenario, where a source transmits private messages to $K$ receivers through a broadcast erasure channel, and the receivers feed back strictly causally and publicly their channel states after each transmission. We explore the achievable rate region when we require that the message to each receiver remains secret - in the information theoretical sense - from all the other receivers. We characterize the capacity of secure communication in all the cases where the capacity of the 1-to-$K$ communication scenario without the requirement of security is known. As a special case, we characterize the secret-message capacity of a single receiver point-to-point erasure channel with public state-feedback in the presence of a passive eavesdropper. We find that in all cases where we have an exact characterization, we can achieve the capacity by using linear complexity two-phase schemes: in the first phase we create appropriate secret keys, and in the second phase we use them to encrypt each message. We find that the amount of key we need is smaller than the size of the message, and equal to the amount of encrypted message the potential eavesdroppers jointly collect. Moreover, we prove that a dishonest receiver that provides deceptive feedback cannot diminish the rate experienced by the honest receivers. We also develop a converse proof which reflects the two-phase structure of our achievability scheme. As a side result, our technique leads to a new outer bound proof for the non-secure communication problem

Secure Communication in Erasure Networks with State-feedback

The security and efficiency of communication are two of the main concerns for networks of today and the future. Our understanding of how to efficiently send information over various channels and networks has significantly increased in the past decade (see e.g., [1–3]), whereas our understanding of how to securely send information has not yet reached the same level. In this thesis, we advance the theory of secure communication by deriving capacity results and by developing coding schemes that provide information-theoretic security for erasure networks. We characterize the highest achievable secret-message rate in the presence of an eavesdropping adversary in various settings, where communication takes place over erasure channels with state-feedback. Our results provide such a characterization for a point-to-point erasure channel, for a broadcast erasure channel with multiple receivers, for a network with multiple parallel channels, a V-network and for a triangle network. We introduce several two-phase secure coding schemes that consist of a key generation phase and an encrypted message sending phase. Our schemes leverage several resources for security: channel erasures, feedback, common randomness and the topology of the network. We present coding schemes for all the above mentioned settings as well as for erasure networks with arbitrary topology. In all the cases where we provide exact characterization, a two-phase scheme achieves the secret-message capacity. All our proposed coding schemes use only linear operations and thus can serve as a basis for practical code designs. For networks, we develop a linear programming framework for describing secure coding schemes and for deriving new outer bounds. We use linear programs to describe our schemes and to prove their optimality. We derive new information theoretic outer bounds. In our intuitive interpretation, our proofs find the connection between the rate of the message and the rate of a secret key that is required to secure the message. Our results reveal nontrivial characteristics of secure communication in erasure networks. We find that – in contrast to non-secure communication – the secret message capacity of a cut does not simplify to the sum of the capacities of the channels that form the cut, moreover, the secret message capacity of a network does not simplify to the minimum secret message capacity of its cuts