Distributed Channel Synthesis

Two familiar notions of correlation are rediscovered as the extreme operating points for distributed synthesis of a discrete memoryless channel, in which a stochastic channel output is generated based on a compressed description of the channel input. Wyner's common information is the minimum description rate needed. However, when common randomness independent of the input is available, the necessary description rate reduces to Shannon's mutual information. This work characterizes the optimal trade-off between the amount of common randomness used and the required rate of description. We also include a number of related derivations, including the effect of limited local randomness, rate requirements for secrecy, applications to game theory, and new insights into common information duality. Our proof makes use of a soft covering lemma, known in the literature for its role in quantifying the resolvability of a channel. The direct proof (achievability) constructs a feasible joint distribution over all parts of the system using a soft covering, from which the behavior of the encoder and decoder is inferred, with no explicit reference to joint typicality or binning. Of auxiliary interest, this work also generalizes and strengthens this soft covering tool.Comment: To appear in IEEE Trans. on Information Theory (submitted Aug., 2012, accepted July, 2013), 26 pages, using IEEEtran.cl

A Stronger Soft-Covering Lemma and Applications

Wyner's soft-covering lemma is a valuable tool for achievability proofs of information theoretic security, resolvability, channel synthesis, and source coding. The result herein sharpens the claim of soft-covering by moving away from an expected value analysis. Instead, a random codebook is shown to achieve the soft-covering phenomenon with high probability. The probability of failure is doubly-exponentially small in the block-length, enabling more powerful applications through the union bound.Comment: IEEE CNS 2015, 2nd Workshop on Physical-layer Methods for Wireless Security, 4 page

Optimal Equivocation in Secrecy Systems a Special Case of Distortion-based Characterization

Recent work characterizing the optimal performance of secrecy systems has made use of a distortion-like metric for partial secrecy as a replacement for the more traditional metric of equivocation. In this work we use the log-loss function to show that the optimal performance limits characterized by equivocation are, in fact, special cases of distortion-based counterparts. This observation illuminates why equivocation doesn't tell the whole story of secrecy. It also justifies the causal-disclosure framework for secrecy (past source symbols and actions revealed to the eavesdropper).Comment: Invited to ITA 2013, 3 pages, no figures, using IEEEtran.cl

Secure Cascade Channel Synthesis

We investigate channel synthesis in a cascade setting where nature provides an iid sequence $X^n$ at node 1. Node 1 can send a message at rate $R_1$ to node 2 and node 2 can send a message at rate $R_2$ to node 3. Additionally, all 3 nodes share bits of common randomness at rate $R_0$. We want to generate sequences $Y^n$ and $Z^n$ along nodes in the cascade such that $(X^n,Y^n,Z^n)$ appears to be appropriately correlated and iid even to an eavesdropper who is cognizant of the messages being sent. We characterize the optimal tradeoff between the amount of common randomness used and the required rates of communication. We also solve the problem for arbitrarily long cascades and provide an inner bound for cascade channel synthesis without an eavesdropper.Comment: ISIT 2013, 5 pages, uses IEEEtran.cl

Gaussian Secure Source Coding and Wyner's Common Information

We study secure source-coding with causal disclosure, under the Gaussian distribution. The optimality of Gaussian auxiliary random variables is shown in various scenarios. We explicitly characterize the tradeoff between the rates of communication and secret key. This tradeoff is the result of a mutual information optimization under Markov constraints. As a corollary, we deduce a general formula for Wyner's Common Information in the Gaussian setting.Comment: ISIT 2015, 5 pages, uses IEEEtran.cl

Rate-Distortion Theory for Secrecy Systems

Secrecy in communication systems is measured herein by the distortion that an adversary incurs. The transmitter and receiver share secret key, which they use to encrypt communication and ensure distortion at an adversary. A model is considered in which an adversary not only intercepts the communication from the transmitter to the receiver, but also potentially has side information. Specifically, the adversary may have causal or noncausal access to a signal that is correlated with the source sequence or the receiver's reconstruction sequence. The main contribution is the characterization of the optimal tradeoff among communication rate, secret key rate, distortion at the adversary, and distortion at the legitimate receiver. It is demonstrated that causal side information at the adversary plays a pivotal role in this tradeoff. It is also shown that measures of secrecy based on normalized equivocation are a special case of the framework.Comment: Update version, to appear in IEEE Transactions on Information Theor

Secure Cascade Channel Synthesis

We consider the problem of generating correlated random variables in a distributed fashion, where communication is constrained to a cascade network. The first node in the cascade observes an i.i.d. sequence $X^n$ locally before initiating communication along the cascade. All nodes share bits of common randomness that are independent of $X^n$. We consider secure synthesis - random variables produced by the system appear to be appropriately correlated and i.i.d. even to an eavesdropper who is cognizant of the communication transmissions. We characterize the optimal tradeoff between the amount of common randomness used and the required rates of communication. We find that not only does common randomness help, its usage exceeds the communication rate requirements. The most efficient scheme is based on a superposition codebook, with the first node selecting messages for all downstream nodes. We also provide a fleeting view of related problems, demonstrating how the optimal rate region may shrink or expand.Comment: Submitted to IEEE Transactions on Information Theor

Tenancy Default, Excess Demand and the Rental Market

We develop a model of a competitive rental housing market with an endogenous rate of tenancy default arising from income uncertainty. Potential tenants must choose to engage in a costly search for rental housing, and must commit to a rental agreement before the uncertainty is resolved. We show that there are two possible equilibria in this market: a market-clearing equilibrium and an equilibrium with excess demand. Therefore, individuals might not have access to rental housing because they are unable to afford to look for housing, they are unable to pay their rent, or with excess demand in the market they are simply unable to find a rental unit. We show that government regulations affecting the cost of default to the housing suppliers and the quality of rental units can have different effects on the equilibrium variables of interest — rental rate, quantity demanded and supplied, and access to rental housing — depending on the type of equilibria in the market. A numerical example illustrates these results.Tenancy Default, Excess Demand, Rental Housing Policies

Equilibrium Excess Demand in the Rental Housing Market (revised)

We develop a model of a competitive rental housing market with endogenous default due to income uncertainty. There is a large number of identical, potential suppliers who each face a fixed cost of entering the rental housing market. Those suppliers who choose to enter decide how many rental units to supply and the rental price to charge. Potential tenants who differ in their income and face an uninsurable income shock choose whether to engage in a costly search for rental housing. If they find a rental unit, then they must commit to a rental agreement before the income uncertainty is resolved. Consequently, some tenants may default on their rental payments. We show that tenancy default can explain persistent excess demand in the rental housing market without any government price regulations. With excess demand in equilibrium, some individuals are simply unable to find rental housing. We study both government regulations affecting the cost of default to the housing suppliers and the quality of rental units, and the imposition of rent control. We show that rent control can have non-standard effects on the access to rental housing and on welfare.Tenancy Default, Excess Demand, Rental Housing Policies