The Secrecy Problems In Multi-user Information Theory

The most important issues in communication are reliability and security. The reliabilityquantifies the maximum rate achievable with small probability of error. Security is an impor-tant issue when the transmitted information is confidential and needs to be kept as secret aspossible from wiretapper. In this thesis, we investigate the reliability and security problemsin multi-user information theory.For the reliability, we investigate the situation that the multiple-access channel (MAC)is controlled by a channel state information sequence, and meanwhile it is available to thechannel encoders in a causal or noncausal manner. This new model can be viewed as aMAC extension of Shannon’s model and Gel’fand-Pinsker’s model. The capacity region isdetermined for the new model in both causal and noncausal manners, and an example abouthow to calculate the capacity region is given.For the security, firstly, we investigate the wiretap channel with memoryless side in-formation. In the new model, the conditional transition probability distribution of the mainchannel depends on a channel state information, which is available at the joint source-channelencoder in a memoryless manner. The capacity region considering transmission rate andequivocation, is determined for the new model. For the causal and noncausal manners, wederive the inner and outer bounds on the capacity region. Furthermore, the secrecy capacityof the new model is described and bounded, which provides the best transmission rate withperfect secrecy.Secondly, the model of wiretap channel has been reconsidered for the case that the mainchannel is controlled by channel state information (side information), and it is available at thetransmitter in a causal case (termed here causal side information) or noncausal case (termedhere noncausal side information). Moreover, there is a noiseless feedback from the legitimatereceiver to the transmitter, and it helps them share a secret key, which enlarges the wiretap-per’s uncertainty about the transmitted message. Measuring the uncertainty by equivocation(conditional entropy), the capacity region considering transmission rate and equivocation, isdetermined for the new model of wiretap channel with causal side information and noiselessfeedback. Furthermore, the secrecy capacity is formulated and bounded. Thirdly, we study the model of wiretap channel with noiseless feedback, and the capac-ity region considering transmission rate and equivocation, is determined. Furthermore, thesecrecy capacity of this model is formulated, which provides the best transmission rate withperfect secrecy. Although the traditional channel capacity is kept in the wiretap channel withnoiseless feedback, the secrecy capacity is larger than that of the model of wiretap channel(without feedback).Fourthly, we study the non-degraded wiretap channel with noiseless feedback, and it isfirst investigated by R. Ahlswede and N. Cai, where a lower and upper bound on the secrecycapacity is provided in their work. However, the capacity region considering transmissionrate and the equivocation to the wiretapper, has not been determined yet. In this article, thecapacity region is determined for the non-degraded wiretap channel with noiseless feedback.Furthermore, the secrecy capacity of this model is formulated.Finally, we investigate the model of degraded broadcast channel with side informa-tion, confidential messages and noiseless feedback. This work is from Steinberg’s work onthe degraded broadcast channel with causal and noncausal side information, and Csisz′arand Ko¨rner’s work on broadcast channel with confidential messages. In this new model,the transmitter sends a confidential message to the non-degraded receiver, and meanwhilesends a common message to both the degraded and non-degraded receivers. Moreover, thechannel for the non-degraded receiver is controlled by channel state information (side infor-mation), and it is available to the transmitter in a causal manner (termed here causal sideinformation) or noncausal manner (termed here noncausal side information). In addition,we assume that there is a noiseless feedback from the output of the channel for the non-degraded receiver to the transmitter, and it helps them share a secret key, which enlarges thedegraded receiver’s uncertainty about the confidential message. Measuring the uncertaintyby equivocation (a conditional entropy), the capacity region composed of all achievable rates-equivocation triples is determined for this new model in both causal and noncausal manners.Furthermore, the secrecy capacity in both manners is formulated.