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arXiv:1701.07380

On the Deterministic Sum-Capacity of the Multiple Access Wiretap Channel

Rick Fritschek, Gerhard Wunder

Published Jan 25, 2017• 1 citation
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Abstract

We study a deterministic approximation of the two-user multiple access wiretap channel. This approximation enables results beyond the recently shown $\tfrac{2}{3}$ secure degrees of freedom (s.d.o.f.) for the Gaussian multiple access channel. While the s.d.o.f. were obtained by real interference alignment, our approach uses signal-scale alignment. We show an achievable scheme which is independent of the rationality of the channel gains. Moreover, our result can differentiate between channel strengths, in particular between both users, and establishes a secrecy rate dependent on this difference. We can show that the resulting achievable secrecy rate tends to the s.d.o.f. for vanishing channel gain differences. Moreover, we extend the s.d.o.f. bound towards a general bound for varying channel strengths and show that our achievable scheme reaches the bound for certain channel gain parameters. We believe that our analysis is the first step towards a constant-gap analysis of the Gaussian multiple access wiretap channel.

DOI
10.48550/arxiv.1701.07380

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We study a deterministic approximation of the two-user multiple access wiretap channel. This approximation enables results beyond the recently shown 23\tfrac{2}{3}32​ secure degrees of freedom (s.d.o.f.) for the Gaussian multiple access channel. While the s.d.o.f. were obtained by real interference alignment, our approach uses signal-scale alignment. We show an achievable scheme which is independent of the rationality of the channel gains. Moreover, our result can differentiate between channel strengths, in particular between both users, and establishes a secrecy rate dependent on this difference. We can show that the resulting achievable secrecy rate tends to the s.d.o.f. for vanishing channel gain differences. Moreover, we extend the s.d.o.f. bound towards a general bound for varying channel strengths and show that our achievable scheme reaches the bound for certain channel gain parameters. We believe that our analysis is the first step towards a constant-gap analysis of the Gaussian multiple access wiretap channel.
We study deterministic approximations of the Gaussian two-user multiple access wiretap channel (G-MAC-WT) and the Gaussian wiretap channel with a helper (G-WT-H). These approximations enable results beyond the recently shown 2/3 and 1/2 secure degrees of freedom (s.d.o.f.) for the G-MAC-WT and the G-WT-H, respectively. While the s.d.o.f. were obtained by real interference alignment, our approach uses signal-scale alignment. We show achievable schemes which are independent of the rationality of the channel gains. Moreover, our results can differentiate between channel strengths, in particular, between both users, and will establish secrecy rates dependent on this difference. We can show that the resulting achievable secrecy rates tend to the s.d.o.f. for vanishing channel gain differences. Moreover, we extend previous and develop new techniques to prove generalized s.d.o.f. bounds for varying channel strengths and show that our achievable schemes reach the bounds for certain channel gain parameters. We believe that our analysis is the next step toward a constant-gap analysis of the G-MAC-WT and the G-WT-H.
Recent investigations have shown that the sum secure degrees of freedom of the Gaussian wiretap channel with a helper is 1/2. The achievable scheme for this result is based on the real interference alignment approach. While providing a good way to show degrees of freedom results, this technique has the disadvantage of relying on the Khintchine-Groshev theorem and is therefore limited to almost all channel gains. This means that there are infinitely many channel gains, where the scheme fails. Furthermore, the real interference alignment approach cannot be used to yield stronger constant-gap results. We approach this topic from a signal-scale alignment perspective and use the linear deterministic model as a first approximation. Here we can show a constant-gap sum capacity for certain channel gain parameters. We transfer these results to the Gaussian model and discuss the results.
We consider a 2-user Gaussian Multiple-Access Wiretap channel (MA-WC), with the unintended receiver having access to signal from only one of the two transmitters. We have characterized the sum-rate capacity of this channel within .5 bits and proved that Gaussian signalling at the transmitters is approximate sum-rate capacity optimal. For the scenario when the additive noise plus the signal from first transmitter have a combined strength poorer than the additive noise of the evesdropper, the above sum-rate capacity characterization leads to an approximate capacity region characterization for the channel. It is shown that, under this circumstances, a timesharing scheme between the two users can achieve the capacity region of the channel within .25 bits. It has been proved (Xie and Ulukus, ISIT 2013 [1]) recently that on a Multiple-Access Wiretap channel interference alignment (IA) is necessary to achieve optimal performance. Our channel model is an interesting variation of the MA-WC for which IA is not necessary and independent Gaussina signalling is sufficient to achieve the optimal sum-rate within .5 bits.
We consider a 2-user Gaussian Multiple-Access Wiretap channel (GMAC-WT), with the Eavesdropper (E) having access to signal from only one (say, T2) of the two transmitters. We characterize the capacity region of this channel approximately within. 5 bits, where the approximation is in terms of only T1's rate or the sum-rate depending on the relative strength of the eavesdropper's channel. However, the approximation is. 5 bits independent of channel coefficients or operating Signal-to-Noise Ratios (SNR). To prove this approximate result we propose two different coding schemes, namely, the power adaptation and the time sharing coding schemes and derive their corresponding achievable rate regions. Both of them use Gaussian input distribution. To establish the approximate capacity, we first derive supersets to the capacity region and then show that corresponding to each rate pair at the boundary of these supersets there exists an achievable rate pair in one of the aforementioned achievable rate regions which are within 1/2 bits to the former pair. In comparison to a very recent result (Xie and Ulukus, ISIT 2013) on a GMAC-WT showing the requirement of interference alignment (IA) to achieve even the degrees of freedom performance, the result of this paper is surprising: our channel model is an interesting variation of the GMAC-WT for which IA is not necessary and Gaussian signalling is sufficient to achieve the entire capacity region within. 5 bits.
In this work, we consider a K-user Gaussian wiretap multiple-access channel (GW-MAC) in which each transmitter has an independent confidential message for the receiver. There is also an external eavesdropper who intercepts the communications. The goal is to transmit the messages reliably while keeping them confidential from the eavesdropper. To accomplish this goal, two different approaches have been proposed in prior works, namely, i.i.d. Gaussian random coding and real alignment. However, the former approach fails at moderate and high SNR regimes as its achievable result does not grow with SNR. On the other hand, while the latter approach gives a promising result at the infinite SNR regime, its extension to the finite-SNR regime is a challenging task. To fill the gap between the performance of the existing approaches, in this work, we establish a new scheme in which, at the receiver's side, it utilizes an extension of the compute-and-forward decoding strategy and at the transmitters' side it exploits lattice alignment, cooperative jamming, and i.i.d. random codes. For the proposed scheme, we derive a new achievable bound on sum secure rate which scales with log(SNR) and hence it outperforms the i.i.d. Gaussian codes in moderate and high SNR regimes. We evaluate the performance of our scheme, both theoretically and numerically. Furthermore, we show that our sum secure rate achieves the optimal sum secure degrees of freedom in the infinite-SNR regime.