Secret Key Capacity For Multipleaccess Channel With Public Feedback
Himanshu Tyagi, Shun Watanabe
备注：
This paper was presented at Allerton 2013. We were hoping to extend the results further, but it is clear that this is not going to happen soon. Now is as good a time as any for sharing this on arXiv. The results will be of interest for anyone working on secret key agreement or common randomness generation
We consider the generation of a secret key (SK) by the inputs and the output of a secure multipleaccess channel (MAC) that additionally have access to a noiseless public communication channel. Under specific restrictions on the protocols, we derive various upper bounds on the rate of such SKs. Specifically, if the public communication consists of only the feedback from the output terminal, then the rate of SKs that can be generated is bounded above by the maximum symmetric rate $R_f^\ast$ in the capacity region of the MAC with feedback. On the other hand, if the public communication is allowed only before and after the transmission over the MAC, then the rate of SKs is bounded above by the maximum symmetric rate $R^\ast$ in the capacity region of the MAC without feedback. Furthermore, for a symmetric MAC, we present a scheme that generates an SK of rate $R_f^\ast$, improving the best previously known achievable rate $R^\ast$. An application of our results establishes the SK capacity for adder MAC, without any restriction on the protocols.
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This paper considers the problem of secret-key agreement with public discussion subject to a peak power constraint $A$ on the channel input. The optimal input distribution is proved to be discrete with finite support. The result is obtained by first transforming the secret-key channel model into an equivalent Gaussian wiretap channel with better noise statistics at the legitimate receiver and then using the fact that the optimal distribution of the Gaussian wiretap channel is discrete. To overcome the computationally heavy search for the optimal discrete distribution, several suboptimal scheme…
We investigate the secret key agreement from correlated vector Gaussian sources in which the legitimate parties can use the public communication with limited rate. For the class of protocols with the one-way public communication, we show that the optimal trade-off between the rate of key generation and the rate of the public communication is characterized as an optimization problem of a Gaussian random variable. The characterization is derived by using the enhancement technique introduced by Weingarten et.al. for MIMO Gaussian broadcast channel.
In this paper, we consider the problem of secret key generation with one-way communication through both a rate-limited public channel and a rate-limited secure channels where the public channel is from Alice to Bob and Eve and the secure channel is from Alice to Bob. In this model, we do not pose any constraints on the sources, i.e. Bob is not degraded to or less noisy than Eve. We obtain the optimal secret key rate in this problem, both for the discrete memoryless sources and vector Gaussian sources. The vector Gaussian characterization is derived by suitably applying the enhancement argument…
Secret-key agreement protocols over wiretap channels controlled by a state parameter are studied. The entire state sequence is known (non-causally) to the sender but not to the receiver and the eavesdropper. Upper and lower bounds on the secret-key capacity are established both with and without public discussion. The proposed coding scheme involves constructing a codebook to create common reconstruction of the state sequence at the sender and the receiver and another secret-key codebook constructed by random binning. For the special case of Gaussian channels, with no public discussion, - the s…
We consider a model for secrecy generation, with three terminals, by means of public interterminal communication, and examine the problem of characterizing all the rates at which all three terminals can generate a ``secret key,'' and -- simultaneously -- two designated terminals can generate a ``private key'' which is effectively concealed from the remaining terminal; both keys are also concealed from an eavesdropper that observes the public communication. Inner and outer bounds for the ``secret key--private key capacity region'' are derived. Under a certain special condition, these bounds coi…
We explore the duality between the simulation and extraction of secret correlations in light of a similar well-known operational duality between the two notions of common information due to Wyner, and Gács and Körner. For the inverse problem of simulating a tripartite noisy correlation from noiseless secret key and unlimited public communication, we show that Winter's (2005) result for the key cost in terms of a conditional version of Wyner's common information can be simply reexpressed in terms of the existence of a bipartite protocol monotone. For the forward problem of key distillation from…
We consider a non-asymptotic (one-shot) version of the multiterminal secret key agreement problem on a finite linear source model. In this model, the observation of each terminal is a linear function of an underlying random vector composed of finitely many i.i.d. uniform random variables. Restricting the public discussion to be a linear function of the terminals' observations, we obtain a characterization of the communication complexity (minimum number of symbols of public discussion) of generating a secret key of maximum length. The minimum discussion is achieved by a non-interactive protocol…
Hypergraphical sources are a natural class of sources for secret key generation, within which different subsets of terminals sharing secrets are allowed to discuss publicly in order to agree upon a global secret key. While their secrecy capacity, i.e., the maximum rate of a secret key that can be agreed upon by the entire set of terminals, is well-understood, what remains open is the maximum rate of a secret key that can be generated when there is a restriction on the overall rate of public discussion allowed. In this work, we obtain a family of explicitly computable upper bounds on the number…
We consider secret key generation for a "pairwise independent network" model in which every pair of terminals observes correlated sources that are independent of sources observed by all other pairs of terminals. The terminals are then allowed to communicate publicly with all such communication being observed by all the terminals. The objective is to generate a secret key shared by a given subset of terminals at the largest rate possible, with the cooperation of any remaining terminals. Secrecy is required from an eavesdropper that has access to the public interterminal communication. A (single…
For the multiterminal secret key agreement problem, new single-letter lower bounds are obtained on the public discussion rate required to achieve any given secret key rate below the secrecy capacity. The results apply to general source model without helpers or wiretapper's side information but can be strengthened for hypergraphical sources. In particular, for the pairwise independent network, the results give rise to a complete characterization of the maximum secret key rate achievable under a constraint on the total discussion rate.
The problem of simultaneously generating a secret key (SK) and private key (PK) pair among three terminals via public discussion is investigated, in which each terminal observes a component of correlated sources. All three terminals are required to generate a common secret key concealed from an eavesdropper that has access to public discussion, while two designated terminals are required to generate an extra private key concealed from both the eavesdropper and the remaining terminal. An outer bound on the SK-PK capacity region was established in [1], and was shown to be achievable for one case…
We study the generation of a secret key of maximum rate by a pair of terminals observing correlated sources and with the means to communicate over a noiseless public com- munication channel. Our main result establishes a structural equivalence between the generation of a maximum rate secret key and the generation of a common randomness that renders the observations of the two terminals conditionally independent. The minimum rate of such common randomness, termed interactive common information, is related to Wyner's notion of common information, and serves to characterize the minimum rate of in…
Practical implementations of secret-key generation are often based on sequential strategies, which handle reliability and secrecy in two successive steps, called reconciliation and privacy amplification. In this paper, we propose an alternative approach based on polar codes that jointly deals with reliability and secrecy. Specifically, we propose secret-key capacity-achieving polar coding schemes for the following models: (i) the degraded binary memoryless source (DBMS) model with rate-unlimited public communication, (ii) the DBMS model with one-way rate-limited public communication, (iii) the…
This paper investigates the secret key generation in the multiterminal source model, where users observing correlated sources discuss interactively under limited rates to agree on a secret key. We focus on a class of sources representable by minimally connected hypergraphs. For such sources, we give a single-letter explicit characterization of the region of achievable secret key rate and public discussion rate tuple. This is the first result that completely characterizes the achievable rate region for a multiterminal source model, which is beyond the PIN model on a tree. We also obtain an expl…
Information theoretic secret key agreement is impossible without making initial assumptions. One type of initial assumption is correlated random variables that are generated by using a noisy channel that connects the terminals. Terminals use the correlated random variables and communication over a reliable public channel to arrive at a shared secret key. Previous channel models assume that each terminal either controls one input to the channel, or receives one output variable of the channel. In this paper, we propose a new channel model of transceivers where each terminal simultaneously contro…
The multiterminal secret key agreement problem by public discussion is formulated with an additional source compression step where, prior to the public discussion phase, users independently compress their private sources to filter out strongly correlated components for generating a common secret key. The objective is to maximize the achievable key rate as a function of the joint entropy of the compressed sources. Since the maximum achievable key rate captures the total amount of information mutual to the compressed sources, an optimal compression scheme essentially maximizes the multivariate m…
We consider a coded cooperative data exchange problem with the goal of generating a secret key. Specifically, we investigate the number of public transmissions required for a set of clients to agree on a secret key with probability one, subject to the constraint that it remains private from an eavesdropper. Although the problems are closely related, we prove that secret key generation with fewest number of linear transmissions is NP-hard, while it is known that the analogous problem in traditional cooperative data exchange can be solved in polynomial time. In doing this, we completely characte…
We investigate the problem of covert and secret key generation over a state-dependent discrete memoryless channel with one-way public discussion in which an adversary, the warden, may arbitrarily choose the channel state. We develop an adaptive protocol that, under conditions that we explicitly specify, not only allows the transmitter and the legitimate receiver to exchange a secret key but also conceals from the active warden whether the protocol is being run. When specialized to passive adversaries that do not control the channel state, we partially characterize the covert secret key capacit…
This paper studies an information-theoretic one-shot variable-length secret key agreement problem with public discussion. Let $X$ and $Y$ be jointly distributed random variables, each taking values in some measurable space. Alice and Bob observe $X$ and $Y$ respectively, can communicate interactively through a public noiseless channel, and want to agree on a key length and a key that is approximately uniformly distributed over all bit sequences with the agreed key length. The public discussion is observed by an eavesdropper, Eve. The key should be approximately independent of the public discus…
We consider distillation of secret bits from partially secret noisy correlations P_ABE, shared between two honest parties and an eavesdropper. The most studied distillation scenario consists of joint operations on a large number of copies of the distribution (P_ABE)^N, assisted with public communication. Here we consider distillation with only one copy of the distribution, and instead of rates, the 'quality' of the distilled secret bits is optimized, where the 'quality' is quantified by the secret-bit fraction of the result. The secret-bit fraction of a binary distribution is the proportion wh…
We consider information theoretic secret key agreement and secure function computation by multiple parties observing correlated data, with access to an interactive public communication channel. Our main result is an upper bound on the secret key length, which is derived using a reduction of binary hypothesis testing to multiparty secret key agreement. Building on this basic result, we derive new converses for multiparty secret key agreement. Furthermore, we derive converse results for the oblivious transfer problem and the bit commitment problem by relating them to secret key agreement. Finall…
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A source model of key sharing between three users is considered in which each pair of them wishes to agree on a secret key hidden from the remaining user. There are rate-limited public channels for communications between the users. We give an inner bound on the secret key capacity region in this framework. Moreover, we investigate a practical setup in which localization information of the users as the correlated observations are exploited to share pairwise keys between the users. The inner and outer bounds of the key capacity region are analyzed in this setup for the case of i.i.d. Gaussian ob…
For a discrete or a continuous source model, we study the problem of secret-key generation with one round of rate-limited public communication between two legitimate users. Although we do not provide new bounds on the wiretap secret-key (WSK) capacity for the discrete source model, we use an alternative achievability scheme that may be useful for practical applications. As a side result, we conveniently extend known bounds to the case of a continuous source model. Specifically, we consider a sequential key-generation strategy, that implements a rate-limited reconciliation step to handle reliab…
We consider the problem of multiterminal secret key agreement (SKA) in wiretapped source model where terminals have access to samples of correlated random variables from a publicly known joint probability distribution. The adversary has access to a side information variable, that is correlated with terminals' variables. We focus on a special type of terminal variables in this model, known as Tree-PIN, where the relation between variables of the terminals can be represented by a tree. The study of Tree-PIN source model is of practical importance as it can be realized in wireless network environ…
The problem of secret sharing over the Gaussian wiretap channel is considered. A source and a destination intend to share secret information over a Gaussian channel in the presence of a wiretapper who observes the transmission through another Gaussian channel. Two constraints are imposed on the source-to-destination channel; namely, the source can transmit only binary phase shift keyed (BPSK) symbols, and symbol-by-symbol hard-decision quantization is applied to the received symbols of the destination. An error-free public channel is also available for the source and destination to exchange me…
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…
Code-based public-key cryptosystems based on QC-LDPC and QC-MDPC codes are promising post-quantum candidates to replace quantum vulnerable classical alternatives. However, a new type of attacks based on Bob's reactions have recently been introduced and appear to significantly reduce the length of the life of any keypair used in these systems. In this paper we estimate the complexity of all known reaction attacks against QC-LDPC and QC-MDPC code-based variants of the McEliece cryptosystem. We also show how the structure of the secret key and, in particular, the secret code rate affect the compl…
We consider the secret key agreement problem under the multiterminal source model proposed by Csiszár and Narayan. A single-letter characterization of the secrecy capacity is desired but remains unknown except in the extreme case with unlimited public discussion and without wiretapper's side information. Taking the problem to the opposite extreme by requiring the public discussion rate to be zero asymptotically, we obtain the desired characterization under surprisingly general setting with wiretapper's side information, silent users, trusted and untrusted helpers. An immediate consequence of t…
We revisit the problem of secret key agreement using interactive public communication for two parties and propose a new secret key agreement protocol. The protocol attains the secret key capacity for general observations and attains the second-order asymptotic term in the maximum length of a secret key for independent and identically distributed observations. In contrast to the previously suggested secret key agreement protocols, the proposed protocol uses interactive communication. In fact, the standard one-way communication protocol used prior to this work fails to attain the asymptotic resu…
For the multiterminal secret key agreement problem under a private source model, it is known that the maximum key rate, i.e., the secrecy capacity, can be achieved through communication for omniscience, but the omniscience strategy can be strictly suboptimal in terms of minimizing the public discussion rate. While a single-letter characterization is not known for the minimum discussion rate needed for achieving the secrecy capacity, we derive single-letter lower and upper bounds that yield some simple conditions for omniscience to be discussion-rate optimal. These conditions turn out to be eno…
Secret sharing over the fast-fading MIMO wiretap channel is considered. A source and a destination try to share secret information over a fast-fading MIMO channel in the presence of a wiretapper who also makes channel observations that are different from but correlated to those made by the destination. An interactive authenticated unrestricted public channel is also available for use by the source and destination in the secret sharing process. This falls under the "channel-type model with wiretapper" considered by Ahlswede and Csiszar. A minor extension of their result (to continuous channel a…
Communication complexity refers to the minimum rate of public communication required for generating a maximal-rate secret key (SK) in the multiterminal source model of Csiszar and Narayan. Tyagi recently characterized this communication complexity for a two-terminal system. We extend the ideas in Tyagi's work to derive a lower bound on communication complexity in the general multiterminal setting. In the important special case of the complete graph pairwise independent network (PIN) model, our bound allows us to determine the exact linear communication complexity, i.e., the communication compl…
We study the ensemble performance of biometric authentication systems, based on secret key generation, which work as follows. In the enrollment stage, an individual provides a biometric signal that is mapped into a secret key and a helper message, the former being prepared to become available to the system at a later time (for authentication), and the latter is stored in a public database. When an authorized user requests authentication, claiming his/her identity as one of the subscribers, s/he has to provide a biometric signal again, and then the system, which retrieves also the helper messag…
This paper deals with public-key steganography in the presence of a passive warden. The aim is to hide secret messages within cover-documents without making the warden suspicious, and without any preliminar secret key sharing. Whereas a practical attempt has been already done to provide a solution to this problem, it suffers of poor flexibility (since embedding and decoding steps highly depend on cover-signals statistics) and of little capacity compared to recent data hiding techniques. Using the same framework, this paper explores the use of trellis-coded quantization techniques (TCQ and turb…
We study the capacity of secret-key agreement over a wiretap channel with state parameters. The transmitter communicates to the legitimate receiver and the eavesdropper over a discrete memoryless wiretap channel with a memoryless state sequence. The transmitter and the legitimate receiver generate a shared secret key, that remains secret from the eavesdropper. No public discussion channel is available. The state sequence is known noncausally to the transmitter. We derive lower and upper bounds on the secret-key capacity. The lower bound involves constructing a common state reconstruction seque…
This paper considers secret-key generation between several agents and a base station that observe independent and identically distributed realizations of correlated random variables. Each agent wishes to generate the longest possible individual key with the base station by means of public communication. All keys must be jointly kept secret from all external entities. In this many-to-one secret-key generation setting, it can be shown that the agents can take advantage of a collective protocol to increase the sum-rate of their generated keys. However, when each agent is only interested in maximi…
In this paper we consider the problem of extracting secret key from an eavesdropped source $p_{XYZ}$ at a rate given by the conditional mutual information. We investigate this question under three different scenarios: (i) Alice ($X$) and Bob ($Y$) are unable to communicate but share common randomness with the eavesdropper Eve ($Z$), (ii) Alice and Bob are allowed one-way public communication, and (iii) Alice and Bob are allowed two-way public communication. Distributions having a key rate of the conditional mutual information are precisely those in which a "helping" Eve offers Alice and Bob no…
We consider problems of two-user secret key generation through an intermediate relay. Each user observes correlated source sequences and communicates to the relay over rate-limited noiseless links. The relay processes and broadcasts information to the two users over another rate-limited link. The users then generate a common secret key based on their sources and information from the relay. In the untrusted relay setting, the goal is to establish key agreement between the two users at the highest key rate without leaking information about the key to the relay. We characterize inner and outer bo…
Collins and Popescu realized a powerful analogy between several resources in classical and quantum information theory. The Collins-Popescu analogy states that public classical communication, private classical communication, and secret key interact with one another somewhat similarly to the way that classical communication, quantum communication, and entanglement interact. This paper discusses the information-theoretic treatment of this analogy for the case of noisy quantum channels. We determine a capacity region for a quantum channel interacting with the noiseless resources of public classica…
We consider the secret key generation problem when sources are randomly excited by the sender and there is a noiseless public discussion channel. Our setting is thus similar to recent works on channels with action-dependent states where the channel state may be influenced by some of the parties involved. We derive single-letter expressions for the secret key capacity through a type of source emulation analysis. We also derive lower bounds on the achievable reliability and secrecy exponents, i.e., the exponential rates of decay of the probability of decoding error and of the information leakage…
A new scenario for generating a secret key and two private keys among three Terminals in the presence of an external eavesdropper is considered. Terminals 1, 2 and 3 intend to share a common secret key concealed from the external eavesdropper (Terminal 4) and simultaneously, each of Terminals 1 and 2 intends to share a private key with Terminal 3 while keeping it concealed from each other and from Terminal 4. All four Terminals observe i.i.d. outputs of correlated sources and there is a public channel from Terminal 3 to Terminals 1 and 2. An inner bound of the "secret key-private keys capacity…
A source model for secret key generation between terminals is considered. Two users, namely users 1 and 2, at one side communicate with another user, namely user 3, at the other side via a public channel where three users can observe i.i.d. outputs of correlated sources. Each of users 1 and 2 intends to share a secret key with user 3 where user 1 acts as a wiretapper for user 2 and vice versa. In this model, two situations are considered: communication from users 1 and 2 to user 3 (the forward key strategy) and from user 3 to users 1 and 2 (the backward key strategy). In both situations, the g…
One-way functions are widely used for encrypting the secret in public key cryptography, although they are regarded as plausibly one-way but have not been proven so. Here we discuss the public key cryptosystem based on the system of higher order Diophantine equations. In this system those Diophantine equations are used as public keys for sender and recipient, and sender can recover the secret from the Diophantine equation returned from recipient with a trapdoor. In general the system of Diophantine equations is hard to solve when it is positive-dimensional and it implies the Diophantine equatio…
Harnessing a block-sparse prior to recover signals through underdetermined linear measurements has been extensively shown to allow exact recovery in conditions where classical compressed sensing would provably fail. We exploit this result to propose a novel private communication framework where the secrecy is achieved by transmitting instances of an unidentifiable compressed sensing problem over a public channel. The legitimate receiver can attempt to overcome this ill-posedness by leveraging secret knowledge of a block structure that was used to encode the transmitter's message. We study the …
Composable, unconditional security without a Quantum secret key: public broadcast channels and their conceptualizations, adaptive bit transmission rates, fidelity pruning under wiretaps
We consider the strong secret key (SK) agreement problem for the satellite communication setting, where a satellite chooses a common binary phase shift keying modulated input for three statistically independent additive white Gaussian noise measurement channels whose outputs are observed by two legitimate transceivers (Alice and Bob) and an eavesdropper (Eve), respectively. Legitimate transceivers have access to an authenticated, noiseless, two-way, and public communication link, so they can exchange multiple rounds of public messages to agree on a SK hidden from Eve. Without loss of essential…
We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings $x$ and $y$ is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties, one having $x$ and the complexity profile of the pair and the other one having $y$ and the complexity profile of the pair, can establish via a probabilistic protocol with interaction on a public channel. For $\ell > 2$, the longest shared secret that can be established from a tuple of strings $(x_1, \ldots , x_\ell)$ by $\ell$ parties, each one having one component of the tuple and …
A subset of a set of terminals that observe correlated signals seek to compute a given function of the signals using public communication. It is required that the value of the function be kept secret from an eavesdropper with access to the communication. We show that the function is securely computable if and only if its entropy is less than the "aided secret key" capacity of an associated secrecy generation model, for which a single-letter characterization is provided.
