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IEEE International Conference on Acoustics Speech and Signal Processing

On the Computability of the Secret Key Capacity under Rate Constraints

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

Published Apr 17, 2019• 15 citations
Publisher

Abstract

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.

DOI
10.1109/icassp.2019.8683122

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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 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 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.
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.
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 paper, the secret-key generation based on a compound source is studied where the realization of the source statistic 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 for a finite compound source is derived as a function of the public communication rate constraint. A multi-letter capacity formula is further computed for a finite compound source for the case in which the public communication is unconstrained. Finally, a single-letter capacity formula is derived for a degraded compound source with an arbitrary (possibly infinite) set of source states and a finite set of marginal states.
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 paper, the secret-key generation based on a compound source is studied where the realization of the source statistic 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 for a finite compound source is derived as a function of the public communication rate constraint. A multi-letter capacity formula is further computed for a finite compound source for the case in which the public communication is unconstrained. Finally, a single-letter capacity formula is derived for a degraded compound source with an arbitrary (possibly infinite) set of source states and a finite set of marginal states.
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.
In secret-key generation using a compound source, the actual statistics of the source are unknown to the participants. It is assumed rather that the actual source belongs to a set (compound set) which is known to the participants. The secret-key generation protocol should guarantee in this case reliability and security of the generated secret-key simultaneously for all elements of the compound set. In this paper, secret-key generation based on a three-party compound source is studied in which an eavesdropper's side information is also taken into account and strong secrecy is guaranteed. At the same time, the public communication rate constraint between the legitimate users is part of the secret-key generation protocol. In this setting, the achievable secret-key rates for finite compound sources are first reformulated as a region of secret-key rate versus communication rate constraint pairs. It is shown that this region is in general convex, even if the compound set is infinite. Based on this, the secret-key capacity results are extended to be valid for arbitrary (possibly infinite) compound sources with a finite set of marginals. In this case, the secret-key capacity is completely characterized as a function of the forward communication rate parameter between the legitimate users.