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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Secret Key Capacity and Advantage Distillation Capacity

Jun Muramatsu, Kazuyuki Yoshimura, P. DAVIS

Published Oct 1, 2006• 8 citations
Publisher

Abstract

Secret key agreement is a procedure for agreeing on a secret key by exchanging messages over a public channel when a sender, a legitimate receiver (henceforth referred to as a receiver), and an eavesdropper have access to correlated sources. Maurer [6] defined secret key capacity, which is the least upper bound of the key generation rate of the secret key agreement, and presented an upper and a lower bound for the secret key capacity. The advantage distillation capacity is introduced and it is shown that this quantity equals to the secret key capacity. Naive information theoretical expressions of the secret key capacity and the advantage distillation capacity are also presented. An example of correlated sources, for which an analytic expression of the secret key capacity can be obtained, is also presented.

DOI
10.1093/ietfec/e89-a.10.2589

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Secret Key Capacity and Advantage Distillation Capacity

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Secret key agreement is a procedure for agreeing on a secret key by exchanging messages over a public channel when a sender, a legitimate receiver and an eavesdropper have access to correlated sources. Maurer, IEEE Trans. Inf. Theory 1993 defined secret key capacity, which is the least upper bound of the key generation rate of the secret key agreement, and presented an upper and a lower bound for the secret key capacity. In this paper, the advantage distillation capacity is introduced and it is shown that this quantity equals to the secret key capacity. A naive information theoretical expression of the secret key capacity and the advantage distillation capacity is also presented. An example of correlated sources, for which an analytic expression of the secret key capacity can be obtained, is also presented

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Abstract—We study the secret-key capacity in a joint source-channel coding setup—the terminals are connected over a discrete memoryless channel and have access to side information, modelled as a pair of discrete memoryless source sequences. As our main result, we establish the upper and lower bounds on the secret-key capacity. In the lower bound expression, the equivocation terms of the source and channel components are functionally additive even though the coding scheme generates a single secret-key by jointly taking into account the source and channel equivocations. Our bounds coincide, thus establishing the capacity, when the underlying wiretap channel can be decomposed into a set of independent, parallel, and reversely degraded channels. For the case of parallel Gaussian channels and jointly Gaussian sources we show that Gaussian codebooks achieve the secret-key capacity. In addition, when the eavesdropper also observes a correlated side information sequence, we establish the secret-key capacity when both the source and channel of the eavesdropper are a degraded version of the legitimate receiver. We finally also treat the case when a public discussion channel is available, propose a separation based coding scheme, and establish its optimality when the channel output symbols of the legitimate receiver and eavesdropper are conditionally independent given the input. Index Terms—Information theoretic security, joint sourcechannel coding, public discussion, secret-key agreement, wiretap channel. I.

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Confidentiality of information is a key consideration in many networking applications, including e-commerce, online banking, and intelligence operations. How can information be communicated reliably to the legitimate users, while keeping it secret from eavesdroppers? How does such a secrecy constraint on communication affect the limits on information flow in the network? In this chapter, we study these questions under the information theoretic notion of secrecy, which requires each eavesdropper to obtain essentially no information about the messages sent from knowledge of its received sequence, the channel statistics, and the codebooks used. We investigate two approaches to achieve secure communication. The first is to exploit the statistics of the channel from the sender to the legitimate receivers and the eavesdroppers. We introduce the wiretap channel as a 2-receiver broadcast channel with a legitimate receiver and an eavesdropper, and establish its secrecy capacity, which is the highest achievable secret communication rate. The idea is to design the encoder so that the channel from the sender to the receiver becomes effectively stronger than the channel to the eavesdropper; hence the receiver can recover the message but the eavesdropper cannot. This wiretap coding scheme involves multicoding and randomized encoding. If the channel from the sender to the receiver is weaker than that to the eavesdropper, however, secret communication at a positive rate is not possible. This brings us to the second approach to achieve secret communication, which is to use a secret key shared between the sender and the receiver but unknown to the eavesdropper. We show that the rate of such secret key must be at least as high as the rate of the confidential message. This raises the question of how the sender and the receiver can agree on such a long secret key in the first place. After all, if they had a confidential channel with sufficiently high capacity to communicate the key, then why not use it to communicate the message itself! We show that if the sender and the receiver have access to correlated sources (e.g., through a satellite beaming common randomness to them), then they can still agree on a secret key even when the channel has zero secrecy capacity. We first consider the source model for key agreement, where the sender communicates with the receiver over a noiseless public broadcast channel to generate a secret key from their correlated sources.

Cambridge University Press eBooks•Dec 8, 2011•1 citation
Secret key agreement is a procedure for agreeing on a secret key by exchanging messages over a public channel when a sender, a legitimate receiver (henceforth referred to as a receiver), and an eavesdropper have access to correlated sources. Maurer [6] defined secret key capacity, which is the least upper bound of the key generation rate of the secret key agreement, and presented an upper and a lower bound for the secret key capacity. The advantage distillation capacity is introduced and it is shown that this quantity equals to the secret key capacity. Naive information theoretical expressions of the secret key capacity and the advantage distillation capacity are also presented. An example of correlated sources, for which an analytic expression of the secret key capacity can be obtained, is also presented.
Secret key agreement is a procedure for agreeing on a secret key by exchanging messages over a public channel when a sender, a legitimate receiver and an eavesdropper have access to correlated sources. Maurer, IEEE Trans. Inf. Theory 1993 defined secret key capacity, which is the least upper bound of the key generation rate of the secret key agreement, and presented an upper and a lower bound for the secret key capacity. In this paper, the advantage distillation capacity is introduced and it is shown that this quantity equals to the secret key capacity. A naive information theoretical expression of the secret key capacity and the advantage distillation capacity is also presented. An example of correlated sources, for which an analytic expression of the secret key capacity can be obtained, is also presented
The secret-key agreement problem over wiretap channels controlled by a state parameter is studied. The entire state sequence is known (non-causally) to the sender but not to the receiver and the eavesdropper. Upper and lower bounds on the secret-key capacity are established both with and without public discussion. In absence of public-discussion, our lower bound is strictly better than the best known lower bound for transmitting an independent secret message. This illustrates that the coding schemes for secret-key agreement are substantially different than secret-message transmission. For the special case of Gaussian channels, we establish the secret-key capacity when the legitimate receiver's signal-to-noise-ratio is greater than 0 dB. When a public discussion channel is available between the sender and the receiver, upper and lower bounds on the secret-key capacity are again established. These bounds coincide, yielding the capacity, when then the channels of the receiver and the eavesdropper satisfy an independent noise condition.
Secret-key agreement protocols over wiretap channels controlled by a state parameter are studied. The entire state sequence is known (non-causally) to the sender but not to the receiver and the eavesdropper. Upper and lower bounds on the secret-key capacity are established both with and without public discussion. The proposed coding scheme involves constructing a codebook to create common reconstruction of the state sequence at the sender and the receiver and another secret-key codebook constructed by random binning. For the special case of Gaussian channels, with no public discussion, - the secret-key generation with dirty paper problem, the gap between our bounds is at-most 1/2 bit and the bounds coincide in the high signal-to-noise ratio and high interference-to-noise ratio regimes. In the presence of public discussion our bounds coincide, yielding the capacity, when then the channels of the receiver and the eavesdropper satisfy an in- dependent noise condition.
Abstract—We study the secret-key capacity in a joint source-channel coding setup—the terminals are connected over a discrete memoryless channel and have access to side information, modelled as a pair of discrete memoryless source sequences. As our main result, we establish the upper and lower bounds on the secret-key capacity. In the lower bound expression, the equivocation terms of the source and channel components are functionally additive even though the coding scheme generates a single secret-key by jointly taking into account the source and channel equivocations. Our bounds coincide, thus establishing the capacity, when the underlying wiretap channel can be decomposed into a set of independent, parallel, and reversely degraded channels. For the case of parallel Gaussian channels and jointly Gaussian sources we show that Gaussian codebooks achieve the secret-key capacity. In addition, when the eavesdropper also observes a correlated side information sequence, we establish the secret-key capacity when both the source and channel of the eavesdropper are a degraded version of the legitimate receiver. We finally also treat the case when a public discussion channel is available, propose a separation based coding scheme, and establish its optimality when the channel output symbols of the legitimate receiver and eavesdropper are conditionally independent given the input. Index Terms—Information theoretic security, joint sourcechannel coding, public discussion, secret-key agreement, wiretap channel. I.
Confidentiality of information is a key consideration in many networking applications, including e-commerce, online banking, and intelligence operations. How can information be communicated reliably to the legitimate users, while keeping it secret from eavesdroppers? How does such a secrecy constraint on communication affect the limits on information flow in the network? In this chapter, we study these questions under the information theoretic notion of secrecy, which requires each eavesdropper to obtain essentially no information about the messages sent from knowledge of its received sequence, the channel statistics, and the codebooks used. We investigate two approaches to achieve secure communication. The first is to exploit the statistics of the channel from the sender to the legitimate receivers and the eavesdroppers. We introduce the wiretap channel as a 2-receiver broadcast channel with a legitimate receiver and an eavesdropper, and establish its secrecy capacity, which is the highest achievable secret communication rate. The idea is to design the encoder so that the channel from the sender to the receiver becomes effectively stronger than the channel to the eavesdropper; hence the receiver can recover the message but the eavesdropper cannot. This wiretap coding scheme involves multicoding and randomized encoding. If the channel from the sender to the receiver is weaker than that to the eavesdropper, however, secret communication at a positive rate is not possible. This brings us to the second approach to achieve secret communication, which is to use a secret key shared between the sender and the receiver but unknown to the eavesdropper. We show that the rate of such secret key must be at least as high as the rate of the confidential message. This raises the question of how the sender and the receiver can agree on such a long secret key in the first place. After all, if they had a confidential channel with sufficiently high capacity to communicate the key, then why not use it to communicate the message itself! We show that if the sender and the receiver have access to correlated sources (e.g., through a satellite beaming common randomness to them), then they can still agree on a secret key even when the channel has zero secrecy capacity. We first consider the source model for key agreement, where the sender communicates with the receiver over a noiseless public broadcast channel to generate a secret key from their correlated sources.