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IEEE Transactions on Signal Processing

On the Algorithmic Computability of the Secret Key and Authentication Capacity Under Channel, Storage, and Privacy Leakage Constraints

Holger Boche, Rafael F. Schaefer, Sebastian Baur, H. Vincent Poor

Published Jul 24, 2019• 32 citations
Publisher

Abstract

Secret key generation refers to the problem of generating a common secret key without revealing any information about it to an eavesdropper. All users observe correlated components of a common source and can further use a noisy public channel for discussion, which is open to eavesdroppers. A related problem is that of secure authentication, which has structural similarities and connections to the first problem. For authentication, users need to be enrolled and securely authenticated while minimizing the privacy leakage rate. This paper studies the algorithmic computability of the forward secret key capacity and the secure authentication capacity. For the algorithmic computability, the concept of a Turing machine is used as it provides fundamental performance limits for today's digital computers. In this paper, it is shown that the forward secret key capacity with a noisy public channel is not computable and consequently, there is no algorithm that can simulate or compute the secret key capacity; even if there are no limitations on computational complexity and computing power. On the other hand, if the public channel is noiseless so that there are no rate constraints on the public communication, the secret key capacity is a computable continuous function, which is the strongest form of computability. A similar behavior is subsequently observed for the authentication problem: The secure authentication capacity under storage rate and privacy leakage rate constraints is not computable, while the case without privacy leakage rate constraints is computable.

DOI
10.1109/tsp.2019.2929467

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Secret key generation refers to the problem of generating a common secret key without revealing any information about it to an eavesdropper. All users observe correlated components of a common source and can further use a noisy public channel for discussion, which is open to eavesdroppers. A related problem is that of secure authentication, which has structural similarities and connections to the first problem. For authentication, users need to be enrolled and securely authenticated while minimizing the privacy leakage rate. This paper studies the algorithmic computability of the forward secret key capacity and the secure authentication capacity. For the algorithmic computability, the concept of a Turing machine is used as it provides fundamental performance limits for today's digital computers. In this paper, it is shown that the forward secret key capacity with a noisy public channel is not computable and consequently, there is no algorithm that can simulate or compute the secret key capacity; even if there are no limitations on computational complexity and computing power. On the other hand, if the public channel is noiseless so that there are no rate constraints on the public communication, the secret key capacity is a computable continuous function, which is the strongest form of computability. A similar behavior is subsequently observed for the authentication problem: The secure authentication capacity under storage rate and privacy leakage rate constraints is not computable, while the case without privacy leakage rate constraints is computable.
Secret key generation refers to the problem of generating a common secret key without revealing any information about it to an eaves-dropper. All users observe correlated components of a common source and can further use a rate-limited public channel for discussion which is open to eavesdroppers. This paper studies the Turing computability of the secret key capacity with a single rate-limited public forward transmission. Turing computability provides fundamental performance limits for today’s digital computers. It is shown that the secret key capacity under rate constraints is not Turing computable, and consequently there is no algorithm that can simulate or compute the secret key capacity, even if there are no limitations on computational complexity and computing power. On the other hand, if there are no rate constraints on the forward transmission, the secret key capacity is Turing computable. This shows that restricting the communication rate over the public channel transforms a Turing computable problem into a non-computable problem. To the best of our knowledge, this is the first time that such a phenomenon has been observed.
Secret-key agreement using physical identifiers is a promising security protocol for the authentication of users and devices with small chips, owing to its lightweight security. In the previous studies, the fundamental limits of such a protocol were analyzed, and the results showed that two auxiliary random variables were involved in the capacity region expressions. However, with two auxiliary random variables, it is difficult to directly apply the expressions to derive the computable forms of the capacity regions for certain information sources such as binary and Gaussian sources, which hold importance in practical applications. In this paper, we explore the structure of authentication channels and reveal that for the classes of degraded and less noisy authentication channels, a single auxiliary random variable is sufficient to express the capacity regions. As specific examples, we use the expressions with one auxiliary random variable to derive the computable forms for binary and Gaussian sources. Numerical calculations for the Gaussian case show the trade-off between secret-key and privacy-leakage rates under a given storage rate, which illustrates how the noise in the enrollment phase affects the capacity region.
We analyze the fundamental trade-off of secret key-based authentication systems in the presence of an eavesdropper for correlated Gaussian sources. A complete characterization of the trade-off among secret-key, storage, and privacy-leakage rates of both generated and chosen secret models is provided. One of the main contributions is revealing that unlike the known results for discrete sources, there is no need for the second auxiliary random variable in characterizing the capacity regions for the Gaussian cases. In addition, it is shown that the strong secrecy for secrecy-leakage of the systems can be achieved by an information-spectrum approach, and the parametric expressions (computable forms) of the capacity regions are also derived.
Secret-key agreement based on biometric or physical identifiers is a promising security protocol for authenticating users or devices with small chips and has been extensively studied recently. Kittichokechai and Caire (2015) investigated the optimal trade-off in a secret-key agreement model with physical identifiers, where the structure of the authentication channels is similar to the wiretap channels, from information theoretic approaches. Later, the model was extended by Günlü et al. (2018) introducing noise in the enrollment phase and cost-constrained actions at the decoder. The results of these studies show that two auxiliary random variables are involved in the expressions of the optimal rate regions of secret-key, storage, and privacy-leakage rates. However, with these two auxiliary random variables, the complexity of computing the rate region may be prohibitively high. Due to this problem, we are interested in exploring classes of authentication channels that need only one auxiliary random variable in the capacity region expression for discrete source settings. The result shows for the class of degraded and less noisy authentication channels, a single auxiliary random variable is sufficient to express the capacity region of the model. As an example, we also derive the capacity region of secret-key, storage, and privacy-leakage rates for binary sources. Furthermore, the capacity region for scalar Gaussian sources is derived under Gaussian authentication channels.
In the classical Secret-Key generation model, Common Randomness is generated by two terminals based on the observation of correlated components of a common source, while keeping it secret from a non-legitimate observer. It is assumed that the statistics of the source are known to all participants. In this work, the Secret-Key generation model based on a compound source is studied where the source realization is unknown. The protocol should guarantee the security and reliability of the generated Secret-Key, simultaneously for all possible realizations of the compound source. A single-letter lower-bound of the Secret-Key capacity is derived for the case where the public communication rate is limited. Furthermore, a multi-letter capacity formula is computed for the case where the public communication is unconstrained.
Shannon theoretic secret key generation by several parties is considered for models in which a secure noisy channel with multiple input and output terminals and a public noiseless channel of unlimited capacity are available for accomplishing this goal. The secret key is generated for a setAof terminals of the noisy channel, with the remaining terminals (if any) cooperating in this task through their public communication. Single-letter lower and upper bounds for secrecy capacities are obtained when secrecy is required from an eavesdropper that observes only the public communication and perhaps also a set of terminals disjoint fromA. These bounds coincide in special cases, but not in general. We also consider models in which different sets of terminals share multiple keys, one for the terminals in each set with secrecy required from the eavesdropper as well as from the terminals not in this set. Partial results include showing links among the associated secrecy capacity region for multiple keys, the transmission capacity region of the multiple access channel defined by the secure noisy channel, and achievable rates for a single secret key for all the terminals.