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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Secrecy Capacity of Wiretap Channels with Additive Colored Gaussian Noise

Hachiro Fujita

Published Jan 1, 2015• 0 citations
Publisher

Abstract

Wyner has shown in his seminal paper on (discrete memoryless) wiretap channels that if the channel between the sender and an eavesdropper is a degraded version of the channel between the sender and the legitimate receiver, then the sender can reliably and securely transmit a message to the receiver, while the eavesdropper obtains absolutely no information about the message. Later, Leung-Yan-Cheong and Hellman extended Wyner's result to the case where the noise is white Gaussian. In this paper we extend the white Gaussian wiretap channel to the colored Gaussian case and show the finite block length secrecy capacity of colored Gaussian wiretap channels. We also show the asymptotic secrecy capacity of a specific colored Gaussian wiretap channel for which optimal power allocation can be found by a water-filling procedure.

DOI
10.1587/transfun.e98.a.1276

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Wyner has shown in his seminal paper on (discrete memoryless) wiretap channels that if the channel between the sender and an eavesdropper is a degraded version of the channel between the sender and the legitimate receiver, then the sender can reliably and securely transmit a message to the receiver, while the eavesdropper obtains absolutely no information about the message. Later, Leung-Yan-Cheong and Hellman extended Wyner's result to the case where the noise is white Gaussian. In this paper we extend the white Gaussian wiretap channel to the colored Gaussian case and show the finite block length secrecy capacity of colored Gaussian wiretap channels. We also show the asymptotic secrecy capacity of a specific colored Gaussian wiretap channel for which optimal power allocation can be found by a water-filling procedure.
Wyner has shown in his seminal paper on (discrete memoryless) wiretap channels that if the channel between the sender and an eavesdropper is a degraded version of the channel between the sender and the legitimate receiver, then the sender can reliably and securely transmit a message to the receiver, while the eavesdropper obtains absolutely no information about the message. Later, Leung-Yan-Cheong and Hellman extended Wyner's result to the case where the noise is white Gaussian. In this paper we extend the white Gaussian wiretap channel to the colored Gaussian case and show the finite block length secrecy capacity of colored Gaussian wiretap channels. We also show an asymptotic lower bound on the secrecy capacity of a specific colored Gaussian wiretap channel for which optimum power allocation can be found by water filling.
The secrecy capacity of an extended communication model of wiretap channelII is determined. In this channel model, the source message is encoded into a digital sequence of length N and transmitted to the legitimate receiver through a discrete memoryless channel (DMC). There exists an eavesdropper who is able to observe arbitrary μ = N α digital symbols from the transmitter through a second DMC, where 0 ≤ α ≤ 1 is a constant real number. A pair of an encoder and a decoder is designed to let the receiver be able to recover the source message with a vanishing decoding error probability and keep the eavesdropper ignorant of the message. This communication model includes a variety of wiretap channels as special cases. The coding scheme is based on that designed by Ozarow and Wyner for the classic wiretap channel II.
The concept of the wiretap channel was first proposed by Wyner. He considered the case in which data is to be transmitted reliably over a discrete memoryless main channel to a legitimate receiver. The wiretapper views the output of the main channel through another discrete memoryless channel. It is assumed that the wiretapper knows the code being used and his only handicap is the additional noise in his signal. The problem is to maximize the transmission rate R to the legitimate receiver and the equivocation d of the wiretapper. In this dissertation, the additive white Gaussian noise wiretap channel is introduced and the set of all achievable (R,d) pairs is determined explicitly through the use of certain special properties of the Gaussian channel. Some useful characterizations of a special class of wiretap channels are also explored. A model of the wiretap channel with feedback is proposed. It turns out that with the introduction of feedback, even when the main channel is inferior to the wiretapper's channel, it is still possible to reliably communicate with the legitimate receiver in complete secrecy. The binary erasure wiretap channel with feedback is examined and inner and outer bounds on the achievable (R,d) region are given. Finally, a scheme for enlarging the capacity region of multiple-access channels using feedback is analyzed. It is shown that conditions under which an enlargement is possible are fairly weak, indicating that feedback can almost increase the capacity region. (Author)
In this paper, the finite-order autoregressive moving average (ARMA) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of signals in both forward and feedback channels. It is shown that the generalized Schalkwijk-Kailath scheme, a capacity-achieving coding scheme for the Gaussian feedback channel, achieves the same maximum rate for the same channel even with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the Gaussian feedback channel. Furthermore, the results are extended to the additive white Gaussian noise (AWGN) channel with quantized feedback. It is shown that the proposed coding scheme achieves a positive secrecy rate. Our result implies that as the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel.
In this paper, the k-th order autoregressive moving average (ARMA(k)) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of the signals in both forward and feedback channels. It is shown that a variant of the generalized Schalkwijk-Kailath scheme, a capacity-achieving coding scheme for the feedback Gaussian channel, achieves the same maximum rate for the same channel with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the feedback channel. Furthermore, the results are extended to the additive white Gaussian noise (AWGN) channel with quantized feedback. It is shown that the proposed coding scheme achieves a positive secrecy rate. As the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel.
Strong secrecy capacity of compound wiretap channels is studied. The known lower bounds for the secrecy capacity of compound finite-state memoryless channels under discrete alphabets are extended to arbitrary uncertainty sets and continuous alphabets under the strong secrecy criterion. The conditions under which these bounds are tight are given. Under the saddle-point condition, the compound secrecy capacity is shown to be equal to that of the worst-case channel. Based on this, the compound Gaussian multiple-input multiple-output wiretap channel is studied under the spectral norm constraint and without the degradedness assumption. First, it is assumed that only the eavesdropper channel is unknown, but is known to have a bounded spectral norm (maximum channel gain). The compound secrecy capacity is established in a closed form and the optimal signaling is identified. The compound capacity equals the worst-case channel capacity and thus establishing the saddlepoint property; the optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is isotropic. The eigenmode power allocation somewhat resembles the standard water-filling but is not identical to it. More general uncertainty sets are considered and the existence of a maximum element is shown to be sufficient for a saddle-point to exist, so that signaling on the worst-case channel achieves the compound capacity of the whole class of channels. The case of rank-constrained eavesdropper is considered and the respective compound secrecy capacity is established. Subsequently, the case of additive uncertainty in the legitimate channel, in addition to the unknown eavesdropper channel, is studied. Its compound secrecy capacity and the optimal signaling are established in a closed form as well, revealing the same saddle-point property. When a saddle-point exists under strong secrecy, strong and weak secrecy compound capacities are equal.