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arXiv:1803.05090

Secret Key Generation from Channel Noise with the Help of a Common Key

Tatsuya Tomaru

Published Mar 14, 2018• 2 citations
arXivHTMLPDF

Abstract

Information-theoretically secure communications are possible when channel noise is usable and when the channel has an intrinsic characteristic that a legitimate receiver (Bob) can use the noise more advantageously than an eavesdropper (Eve). This report deals with the case in which the channel does not have such an intrinsic characteristic. Here, we use a pre-shared common key as a tool that extrinsically makes Bob more advantageous than Eve. This method uses error-correcting code in addition to the common key and noise, and manages the three components in random-number transmission. Secret keys are generated from noise, and messages are encrypted with the secret keys in a one-time pad manner. As a result, information leaks meaningful to Eve are restricted to the parity-check symbols for the random numbers. It is possible to derive the candidates of the common key from the parity check symbols, and the security of this method is quantified in terms of the amount of computations needed for an exhaustive search of the candidates, where we evaluate the security by assuming that all parity check symbols leak to Eve without bit errors. Noise contributes to not only generating secret keys but also enhancing the security because the candidates of the common key increase with it.

DOI
10.48550/arxiv.1803.05090

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Information-theoretically secure communications are possible when channel noise is usable and when the channel has an intrinsic characteristic that a legitimate receiver (Bob) can use the noise more advantageously than an eavesdropper (Eve). This report deals with the case in which the channel does not have such an intrinsic characteristic. Here, we use a pre-shared common key as a tool that extrinsically makes Bob more advantageous than Eve. This method uses error-correcting code in addition to the common key and noise, and manages the three components in random-number transmission. Secret keys are generated from noise, and messages are encrypted with the secret keys in a one-time pad manner. As a result, information leaks meaningful to Eve are restricted to the parity-check symbols for the random numbers. It is possible to derive the candidates of the common key from the parity check symbols, and the security of this method is quantified in terms of the amount of computations needed for an exhaustive search of the candidates, where we evaluate the security by assuming that all parity check symbols leak to Eve without bit errors. Noise contributes to not only generating secret keys but also enhancing the security because the candidates of the common key increase with it.
Two legitimate parties, referred to as Alice and Bob, wish to generate secret keys from the wireless channel in the presence of an eavesdropper, referred to as Eve, in order to use such keys for encryption and decryption. In general, the secret key rate highly depends on the coherence time of the channel. In particular, a straightforward method of generating secret keys in static environments results in ultra-low rates. In order to resolve this problem, we introduce a low-complexity method called induced randomness. In this method, Alice and Bob independently generate local randomness to be used together with the uniqueness of the wireless channel coefficients in order to enable high-rate secret key generation. In this work, two scenarios are considered: first, when Alice and Bob share a direct communication channel, and second, when Alice and Bob do not have a direct link and communicate through an untrusted relay. After exchanging the induced randomness, post-processing is done by Alice and Bob to generate highly-correlated samples that are used for the key generation. Such samples are then converted into bits, disparities between the sequences generated by Alice and Bob are mitigated, and the resulting sequences are then hashed to compensate for the information leakage to the eavesdropper and to allow consistency checking of the generated key bit sequences. We utilize semantic security measures and information-theoretic inequalities to upper bound the probability of successful eavesdropping attack in terms of the mutual information measures that can be numerically computed. Given certain reasonable system parameters this bound is numerically evaluated to be 2-31and 2-10.57in the first and the second scenario, respectively.
The broadcast channel with independent secret keys is studied. In this scenario, a common message has to be securely broadcast to two legitimate receivers in the presence of an eavesdropper. The transmitter shares with each legitimate receiver an independent secret key of arbitrary rate. These keys can either be used as one-time pads to encrypt the common message or can be interpreted as fictitious messages used as randomization resources for wiretap coding. Both approaches are discussed and the secrecy capacity is derived for various cases. Depending on the qualities of the legitimate and eavesdropper channels, either a one-time pad, wiretap coding, or a combination of both turns out to be capacity-achieving.
In Chapter 3, we considered the transmission of information over a noisy broadcast channel subject to reliability and security constraints; we showed that appropriate coding schemes can exploit the presence of noise to confuse the eavesdropper and guarantee some amount of information-theoretic security. It is important to note that the wiretap channel model assumes that all communications occur over the channel, hence communications are inherently rate-limited and one-way. Consequently, the results obtained do not fully capture the role of noise for secrecy; in particular, for situations in which the secrecy capacity is zero, it is not entirely clear whether this stems from the lack of any “physical advantage” over the eavesdropper or the restrictions imposed on the communication schemes. The objective of this chapter is to study more precisely the fundamental role of noise in information-theoretic security. Instead of studying how we can communicate messages securely over a noisy channel, we now analyze how much secrecy we can extract from the noise itself in the form of a secret key. Specifically, we assume that the legitimate parties and the eavesdropper observe the realizations of correlated random variables and that the legitimate parties attempt to agree on a secret key unknown to the eavesdropper. To isolate the role played by noise, we remove restrictions on communication schemes and we assume that the legitimate parties can distill their key by communicating over a two-way, public, noiseless , and authenticated channel at no cost.
Alice and Bob want to share a secret key and to communicate an independent message, both of which they desire to be kept secret from an eavesdropper Eve. We study this problem of secret communication and secret key generation when two resources are available – correlated sources at Alice, Bob, and Eve, and a noisy broadcast channel from Alice to Bob and Eve which is independent of the sources. We are interested in characterizing the fundamental trade-off between the rates of the secret message and secret key. We present an achievable solution and prove its optimality for the parallel channels and sources case when each sub-channel and source component satisfies a degradation order (either in favor of the legitimate receiver or the eavesdropper). This includes the case of jointly Gaussian sources and an additive Gaussian channel, for which the secrecy region is evaluated. I.
In MIMO wireless channels, channel estimation can provide a source of randomness common to sender and receiver that may be used to generate secret keys. These keys can provide information theoretically secure communication. Alternatively, a known MIMO channel allows transmitting artificial noise in the null space of the main channel to secure communication. In this paper, we show that both methods can be combined to further enhance secrecy. Symbols encrypted at the transmitter simultaneously act as message for the intended receiver and as noise for any potential eavesdroppers. We derive expressions for the minimum-guaranteeable secrecy rate as a function of number of symbols encrypted. Given at least one symbol can be encrypted, our proposed method outperforms the standard artificial noise scheme in terms of both secrecy rate and power expenditure. We first analyze the system assuming keys can be instantaneously extracted from channel measurements. We then derive bounds for rates assuming the overhead time required for key generation produces a Gaussian channel-estimation error.
Alice and Bob want to share a secret key and to communicate an independent message, both of which they desire to be kept secret from an eavesdropper Eve. This problem of secret communication and secret-key generation when two resources are available-correlated sources at Alice, Bob, and Eve, and a noisy broadcast channel from Alice to Bob and Eve which is independent of the sources is studied. The goal is to characterize the fundamental tradeoff between the rates of the secret message and secret key. An achievable solution and proof of its optimality for the parallel channels and sources case when each subchannel and source component satisfies a degradation order (either in favor of the legitimate receiver or the eavesdropper) is presented. This includes the case of jointly Gaussian sources and an additive Gaussian channel, for which the secrecy region is evaluated.