The Gaussian Wiretap Channel with Noisy Public Feedback: Breaking the High-SNR Ceiling T` ung T. Kim and H. Vincent Poor Department of Electrical Engineering Princeton University Princeton, NJ 08544, USA Email: {thanhkim, poor}@princeton.edu Abstract—A multiple-antenna Gaussian wiretap channel in which the number of antennas at the source is not larger than that at the eavesdropper is considered. Without feedback, the secrecy capacity over such a channel generally converges to a constant at high signal-to-noise ratio (SNR). A half-duplex secure protocol allowing the destination to actively broadcast random keys over insecure channels is proposed. It is shown that using multiple antennas at the destination is instrumental in achieving a secrecy rate that grows linearly with log SNR. The pre-log factor of the secrecy rate, i.e. the secure degree of freedom, is characterized, revealing an interesting interplay between the numbers of antennas at the three communication nodes. I. I NTRODUCTION Consider a Gaussian wiretap channel in which the number of antennas at the eavesdropper is greater than or equal to the number of antennas at the source. It is known that in such a restrictive scenario, if the destination passively listens to the transmitted signals from the source, then the secrecy capacity converges to a finite constant at high signal-to-noise ratio (SNR); see for example [1]–[3]. If the destination can actively transmit jamming signals, a more encouraging picture emerges: In modulo-additive chan- nels, the source can send at a secrecy rate that equals to the capacity of a channel without any eavesdropper [4]. However a complete characterization of the secrecy capacity in the important case of additive white Gaussian noise (AWGN) channels has not been obtained. In a related line of thought, achievable secrecy rates for Gaussian two-way channels are studied in [5] for full-duplex systems. In the more challenging and arguably more practical case of half-duplex systems, finding secure schemes that can provide secrecy rate that grows linearly in log SNR is still an open problem to date. In the current work we make progress towards settling this important issue. Inspired by the classical concept of self- interference cancelation in two-way channels [6], we propose and analyze a novel half-duplex scheme that can achieve a positive secure degree of freedom (i.e., the pre-log factor of the secrecy rates) even when the eavesdropper has more antennas than the source. An additional advantage of the proposed scheme is its robustness: both the source and the destination can operate without using any explicit knowledge about the eavesdropper’s channel matrices. While they still need to agree on the number of extra codewords to confuse the eavesdropper, this can in principle be accomplished by assuming a secure S D E H SE H SD H DS H DE Fig. 1. System model. zone around the legitimate nodes that the eavesdropper cannot infiltrate [1]. Loosely speaking, the approach in the current work is to use randomly generated keys known only to the destination to hide the secret message in the form of interference. Due to the half- duplex constraint these random keys need to be first broad- cast over the wireless channel, and thus the source and the eavesdropper also have access to noisy versions of these keys. Unfortunately, in the high-SNR regime, the eavesdropper may as well estimate these keys with (increasingly) high accuracy. Our problem therefore has a flavor of secret communications with feedback over an insecure public channel [7]. Our main idea is to make use of multiple antennas at the destination to send key vectors of sufficiently high dimension, so that even at high SNRs the eavesdropper is unable to completely cancel the interference. As long as the number of antennas at the destination is larger than that at the eavesdropper, we show that strictly positive secure degrees of freedom are typically achievable, regardless of the number of antennas at the source. We characterize the exact secure degree of freedom, which depends on a relationship between the channel matrix from the destination to the source and that from the destination to the eavesdropper, The results highlight an interesting interplay between the number of antennas at three communication nodes. II. SYSTEM MODEL Consider the complex baseband of a multiple-antenna Gaus- sian wiretap channel depicted in Fig. 1. The source S, which has N t antennas, wishes to transmit a secure message to the destination D, which has N r antennas, in the presence of 819 978-1-4244-5827-1/09/$26.00 ©2009 IEEE Asilomar 2009