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arXiv:1710.04643 · IEEE Transactions on Information Theory

Secret-Key Generation in Many-to-One Networks: An Integrated Game-Theoretic and Information-Theoretic Approach

Rémi A. Chou, Aylin Yener

Published Feb 27, 2019• 13 citations
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Abstract

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 maximizing its own secret-key rate, agents may be unwilling to participate in a collective protocol. Furthermore, when such a collective protocol is employed, how to fairly allocate individual key rates arises as a valid issue. This paper studies the tension between cooperation and self-interest with a game-theoretic treatment. This paper establishes that cooperation is in the best interest of all individualistic agents and that there exist individual secret-key rate allocations that incentivize the agents to follow the protocol. In addition, an explicit coding scheme that achieves such allocations is proposed.

DOI
10.1109/tit.2019.2902068

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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 maximizing its own secret-key rate, agents may be unwilling to participate in a collective protocol. Furthermore, when such a collective protocol is employed, how to fairly allocate individual key rates arises as a valid issue. This paper studies the tension between cooperation and self-interest with a game-theoretic treatment. This paper establishes that cooperation is in the best interest of all individualistic agents and that there exist individual secret-key rate allocations that incentivize the agents to follow the protocol. In addition, an explicit coding scheme that achieves such allocations is proposed.
We consider secret-key generation between several agents and a base station that observe independent and identically distributed (i.i.d.) 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. We do not require them to be kept secret among the agents. 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 all the generated keys. However, when each agent is only interested in maximizing its own secret-key rate, agents may be unwilling to participate in a collective protocol. Furthermore, when such a collective protocol is employed, how to fairly allocate individual key rates arises as a valid issue. We study this tension between cooperation and self-interest with a game-theoretic treatment. We establish that cooperation is in the best interest of all agents and that there exists individual secret-key rate allocations that incentivize the agents to follow the protocol. Additionally, we propose an explicit and low-complexity coding scheme based on polar codes and hash functions that achieves such allocations.
This paper investigates the secret key generation from multiple cooperative sources. We assume that the public communication between the legitimate users has unlimited capacity. In this case, the secret key rate is the upper bound of a practical system with limited public communication. The random signals from the multiple helpers can help to create correlated Gaussian sources for the legitimate users to generate secret key, and only local channel state information (CSI) is needed by the helpers to cooperate with others. The power allocation between the helpers is studied with a total transmit power constraint, which results in a non-convex optimization problem. A high efficiency sub-optimal solution is proposed. And even when the legitimate users have no idea about the eavesdropper, the scheme still can help the system to achieve fairly good performance.
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-letter) formula for secret key capacity brings out a natural connection between the problem of secret key generation and a combinatorial problem of maximal packing of Steiner trees in an associated multigraph. An explicit algorithm is proposed for secret key generation based on a maximal packing of Steiner trees in a multigraph; the corresponding maximum rate of Steiner tree packing is thus a lower bound for the secret key capacity. When only two of the terminals or when all the terminals seek to share a secret key, the mentioned algorithm achieves secret key capacity in which case the bound is tight.
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-letter) formula for secret key capacity brings out a natural connection between the problem of secret key generation and a combinatorial problem of maximal packing of Steiner trees in an associated multigraph. An explicit algorithm is proposed for secret key generation based on a maximal packing of Steiner trees in a multigraph; the corresponding maximum rate of Steiner tree packing is thus a lower bound for the secret key capacity. When only two of the terminals or when all the terminals seek to share a secret key, the mentioned algorithm achieves secret key capacity in which case the bound is tight.
Secret key generation by multiple terminals is considered based on their observations of jointly distributed Gaussian signals, followed by public communication among themselves. Exploiting an inherent connection between secrecy generation and lossy data compression, two main contributions are made. The first is a characterization of strong secret key capacity, and entails a converse proof technique that is valid for real-valued (and not necessarily Gaussian) as well as finite-valued signals. The capacity formula acquires a simple form when the terminals observe “symmetrically correlated” jointly Gaussian signals. For the latter setup with two terminals, considering schemes that involve quantization at one terminal, the best rate of an achievable secret key is characterized as a function of quantization rate; secret key capacity is attained as the quantization rate tends to infinity. Structured codes are shown to attain the optimum tradeoff between secret key rate and quantization rate, constituting our second main contribution.
We study common randomness where two parties have access to i.i.d. samples from a known random source, and wish to generate a shared random key using limited (or no) communication with the largest possible probability of agreement. This problem is at the core of secret key generation in cryptography, with connections to communication under uncertainty and locality sensitive hashing. We take the approach of treating correlated sources as a critical resource, and ask whether common randomness can be generated resource-efficiently. We consider two notable sources in this setup arising from correlated bits and correlated Gaussians. We design the first explicit schemes that use only a polynomial number of samples (in the key length) so that the players can generate shared keys that agree with constant probability using optimal communication. The best previously known schemes were both non-constructive and used an exponential number of samples. In the amortized setting, we characterize the largest achievable ratio of key length to communication in terms of the external and internal information costs, two well-studied quantities in theoretical computer science. In the relaxed setting where the two parties merely wish to improve the correlation between the generated keys of length k, we show that there are no interactive protocols using o(k) bits of communication having agreement probability even as small as 2–o(k). For the related communication problem where the players wish to compute a joint function f of their inputs using i.i.d samples from a known source, we give a simultaneous message passing protocol using 2O(c) bits where c is the interactive randomized public-coin communication complexity of f. This matches the lower bound shown previously while the best previously known upper bound was doubly exponential in c. Our schemes reveal a new connection between common randomness and unbiased error-correcting codes, e.g., dual-BCH codes and their analogues in Euclidean space.