Research On Some Information Theory Problems Under Multi-user Or Security Constraints

Claude E.Shannon's epochal “A Mathematical Theory of Communication” marks the dawn of information theory.Up to today,information theory has progressed from a single theoretical paper to an important discipline with a broad field.As a tremendous contribution,information theory has promoted the rapid development of the communication theory.Based on Shannon's channel capacity theorem,it has derived a broad and important research branches including network information theory,network coding,error exponent and strong converse,and also contributed numerous impressive results.However,the contribution of information theory is much more.Indeed,information theory has fundamental contributions to make in statistics,probability theory,cryptography,and authentication theory.Therefore,this dissertation studies some theoretical topics in different fields based on information theory.Firstly,in the coded cache network,by considering the case where the error probability asymptotically tends to zero,we study the two-user's caching problem with asymmetric cache sizes and link qualities.By following the symmetry simplifying idea of Tian,we prove the sufficiency and optimality of the file-index-symmetric caching and delivery codes.Furthermore,under the condition of the symmetric codes,the system performance which is measured by the worst-case delivery latency is fully characterized for an arbitrary number of files.Moreover,the optimal results are extended to the caching problem with heterogeneous distortion requirements.Secondly,in the problem of the error exponent,by considering the case in which the error probability exponentially decays to zero,we study the problem of the erasure exponents for the asymmetric broadcast channel.For this problem,we consider an i.i.d.random superposition codebook and the Forney's optimal decoder with erasure option.Furthermore,we derive ensemble-tight total and undetected error exponents based on the type method and the large deviations analysis.It is shown that the optimal decoder designed to decode the pair of messages achieves the optimal trade-off between the total and undetected error exponents associated with the optimal decoder for the private message.Thirdly,for the problem of hypothesis testing with a communication constraint,by considering the case that the error probability is a constant between 0 and 1,we study the strong converse for hypothesis testing against independence over a two-hop network.By combining two recently proposed techniques for proving strong converse theorems,namely the strong converse technique based on functional inequalities and reverse hyper-contractivity of Markov semigroups by Liu,Handel and Verdu,and the strong converse technique by Tyagi and Watanabe in which the authors used a change-of-measure technique and replaced hard Markov constraints with soft information costs,we prove a strong converse for the optimal trade-off between the communication rates and the exponents of type-II error probabilities.Fourthly,in the problem of the key sharing,by considering the case where the error probability asymptotically tends to zero,we study the problem of the secret key generation from correlated sources and secure link.For this problem,we obtain the optimal secret key rate in this problem and show that the optimal scheme is to conduct secret key generation and key distribution jointly.By comparing two schemes,namely the separation-based scheme where the secure channel is used for key distribution,and the “ideal” scheme in which the secure channel is “ideally” used for key generation,it is shown that the optimal scheme can provide more than one bit of secret key rate for every bit in the secret channel under the secrecy constraint.Finally,for the deception problem with side information in biometric authentication system,by considering the case in which the error probability exponentially decays to zero,we study the problem of the deception exponent with side information.For this problem,we find the optimal trade-off between the exponent of the successful deception probability and the distortion constraint of the authentication system by proving both the achievability and the converse.Our proof is based on a dual connection between this problem of deception and the rate distortion problem with side information at both the encoder and the decoder,where the achievability is based on the construction of an achievable deception function from the rate distortion code,and the converse is proved by the contradiction where we construct a contradictory achievability for the rate distortion problem.