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Asilomar Conference on Signals, Systems and Computers (ACSSC)

Stealthy keyless secret key generation from degraded sources

Pin-Hsun Lin, Carsten Rudolf Janda, Eduard Axel Jorswieck, Rafael F. Schaefer

Published Oct 1, 2017• 3 citations
Publisher

Abstract

To conceal the behavior of communications, we may adopt covert or stealthy communications under different scenarios and demands. However, when legitimate users have worse channels than the eavesdropper, the legitimate users may need additional keys to attain the concealment. Normal secret key generation (SKG) schemes may raise Willie's attention due to the public discussion breaking the concealment. Therefore, in this work, we investigate how to conceal the SKG itself, especially according to the stealth constraint. We prove that if the virtual channel between the common random source observed by the legitimate Alice and Bob, namely, X and Y, respectively, is more capable than that between Bob and Willie (the warden), namely, Y and Z, respectively, the same stealthy SK rate as in the case without stealth constraint can be achieved. Furthermore, if (X, Y, Z) form a Markov chain, then the key capacity of the SKG with the stealth constraint can be achieved for free.

DOI
10.1109/acssc.2017.8335126

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To conceal the behavior of communications, we may adopt covert or stealthy communications under different scenarios and demands. However, when legitimate users have worse channels than the eavesdropper, the legitimate users may need additional keys to attain the concealment. Normal secret key generation (SKG) schemes may raise Willie's attention due to the public discussion breaking the concealment. Therefore, in this work, we investigate how to conceal the SKG itself, especially according to the stealth constraint. We prove that if the virtual channel between the common random source observed by the legitimate Alice and Bob, namely, X and Y, respectively, is more capable than that between Bob and Willie (the warden), namely, Y and Z, respectively, the same stealthy SK rate as in the case without stealth constraint can be achieved. Furthermore, if (X, Y, Z) form a Markov chain, then the key capacity of the SKG with the stealth constraint can be achieved for free.
In order to make a warden, Willie, unaware of the existence of meaningful communications, there have been different schemes proposed including covert and stealth communications. When legitimate users have no channel advantage over Willie, the legitimate users may need additional secret keys to confuse Willie, if the stealth or covert communication is still possible. However, secret key generation (SKG) may raise Willie's attention since it has a public discussion, which is observable by Willie. To prevent Willie's attention, we consider the source model for SKG under a strong secrecy constraint, which has further to fulfill a stealth constraint. Our first contribution is that, if the stochastic dependence between the observations at Alice and Bob fulfills the strict more capable criterion with respect to the stochastic dependence between the observations at Alice and Willie or between Bob and Willie, then a positive stealthy secret key rate is identical to the one without the stealth constraint. Our second contribution is that, if the random variables observed at Alice, Bob, and Willie induced by the common random source form a Markov chain, then the key capacity of the source model SKG with the strong secrecy constraint and the stealth constraint is equal to the key capacity with the strong secrecy constraint, but without the stealth constraint. For the case of fast fading models, a sufficient condition for the existence of an equivalent model, which is degraded, is provided, based on stochastic orders. Furthermore, we present an example to illustrate our results.
In order to make a warden, Willie, unaware of the existence of meaningful communications, there have been different schemes proposed including covert and stealth communications. When legitimate users have no channel advantage over Willie, the legitimate users may need additional secret keys to confuse Willie, if the stealth or covert communication is still possible. However, secret key generation (SKG) may raise Willie’s attention since it has a public discussion, which is observable by Willie. To prevent Willie’s attention, we consider the source model for SKG under a strong secrecy constraint, which has further to fulfill a stealth constraint. Our first contribution is that, if the stochastic dependence between the observations at Alice and Bob fulfills the strict more capable criterion with respect to the stochastic dependence between the observations at Alice and Willie or between Bob and Willie, then a positive stealthy secret key rate is identical to the one without the stealth constraint. Our second contribution is that, if the random variables observed at Alice, Bob, and Willie induced by the common random source form a Markov chain, then the key capacity of the source model SKG with the strong secrecy constraint and the stealth constraint is equal to the key capacity with the strong secrecy constraint, but without the stealth constraint. For the case of fast fading models, a sufficient condition for the existence of an equivalent model, which is degraded, is provided, based on stochastic orders. Furthermore, we present an example to illustrate our results.
In this work, we aim to design a stealthy secret key generation (SSKG) scheme for a completely concealed communication system. More specifically, the data transmission is accomplished by covert communications, which requires additional keys to guarantee the covertness when Bob's channel quality is worse than a warden Willie's in a special definition. However, normal SKG schemes may raise Willie's attention and violate the design goal. Therefore, we resort to the SSKG. The derived results show that the SK capacity lower and upper bounds of the source-model SKG are not affected by the additional stealth constraint. This result implies that we can attain the stealthy SKG capacity for free when the common randomness observed at each node forms a Markov chain. We then prove that the sufficient condition to attain the SK capacity can be relaxed from physical to stochastic degradedness. In order to underline the practical relevance, we also derive a sufficient condition to attain the degradedness by the usual stochastic order for the fast fading Gaussian Maurer's (satellite) model for the source of common randomness.
In this work, we consider a complete covert communication system, which includes the source-model of a stealthy secret key generation (SSKG) as the first phase. The generated key will be used for the covert communication in the second phase of the current round and also in the first phase of the next round. We investigate the stealthy SK rate performance of the first phase. The derived results show that the SK capacity lower and upper bounds of the source-model SKG are not affected by the additional stealth constraint. This result implies that we can attain the SSKG capacity for free when the sequences observed by the three terminals Alice (XnX^nXn), Bob (YnY^nYn) and Willie (ZnZ^nZn) follow a Markov chain relationship, i.e., Xn−Yn−ZnX^n-Y^n-Z^nXn−Yn−Zn. We then prove that the sufficient condition to attain both, the SK capacity as well as the SSK capacity, can be relaxed from physical to stochastic degradedness. In order to underline the practical relevance, we also derive a sufficient condition to attain the degradedness by the usual stochastic order for Maurer's fast fading Gaussian (satellite) model for the source of common randomness.
We study covert secret key generation over a binary-input two-user multiple access channel with one-way public discussion and derive bounds on the capacity region. Specifically, in this problem, there are three legitimate parties: Alice, Bob and Charlie. The goal is to allow Charlie to generate a secret key with Alice and another secret key with Bob, reliably, secretly and covertly. Reliability ensures that the key generated by Alice and Charlie is the same and the key generated by Bob and Charlie is the same. Secrecy ensures that the secret keys generated are only known to specific legitimate parties. Covertness ensures that the key generation process is undetectable by a warden Willie. As a corollary of our result, we establish bounds on the capacity region of wiretap secret key generation without the covertness constraint and discuss the impact of covertness. Our results generalize the point-to-point result of Tahmasbi and Bloch (TIFS 2020) to the setting of multiterminal communication.
We study covert secret key generation over a binary-input two-user multiple access channel with one-way public discussion and derive bounds on the capacity region. Specifically, in this problem, there are three legitimate parties: Alice, Bob and Charlie. The goal is to allow Charlie to generate a secret key with Alice and another secret key with Bob, reliably, secretly and covertly. Reliability ensures that the key generated by Alice and Charlie is the same and the key generated by Bob and Charlie is the same. Secrecy ensures that the secret keys generated are only known to specific legitimate parties. Covertness ensures that the key generation process is undetectable by a warden Willie. As a corollary of our result, we establish bounds on the capacity region of wiretap secret key generation without the covertness constraint and discuss the impact of covertness. Our results generalize the point-to-point result of Tahmasbi and Bloch (TIFS 2020) to the setting of multiterminal communication.