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IEEE Military Communications Conference (MILCOM)

Second-Order Analysis of Secret-Key Capacity From a MIMO Channel

Ahmed Maksud, Yingbo Hua

Published Oct 30, 2023• 1 citation
Publisher

Abstract

This paper presents an analysis of Maurer-Ahlswede-Csiszar (MAC) bounds on secret-key capacity (SKC) based on a MIMO channel probing scheme between two multi-antenna nodes against a multi-antenna eavesdropper (Eve). To reveal useful insights into SKC, we consider the regime of high signal powers or high signal-to-noise ratios (SNR) where SKC consists of a first-order term (FoT) which is proportional to the logarithm of signal power, and a second-order term (SoT) which is invariant to signal power. Applying random-matrix theory for large numbers of antennas on all nodes, this paper shows the FoT and SoT of a lower MAC bound on SKC. While the FoT is consistent with prior work, which vanishes if none of the two nodes has more antennas than Eve and the channel is not perfectly reciprocal, the SoT shows a positive secrecy that increases linearly with the length of random transmissions from one node to another during channel probing even if Eve has many more antennas than the legitimate nodes.

DOI
10.1109/milcom58377.2023.10356312

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This paper presents an analysis of Maurer-Ahlswede-Csiszar (MAC) bounds on secret-key capacity (SKC) based on a MIMO channel probing scheme between two multi-antenna nodes against a multi-antenna eavesdropper (Eve). To reveal useful insights into SKC, we consider the regime of high signal powers or high signal-to-noise ratios (SNR) where SKC consists of a first-order term (FoT) which is proportional to the logarithm of signal power, and a second-order term (SoT) which is invariant to signal power. Applying random-matrix theory for large numbers of antennas on all nodes, this paper shows the FoT and SoT of a lower MAC bound on SKC. While the FoT is consistent with prior work, which vanishes if none of the two nodes has more antennas than Eve and the channel is not perfectly reciprocal, the SoT shows a positive secrecy that increases linearly with the length of random transmissions from one node to another during channel probing even if Eve has many more antennas than the legitimate nodes.
This paper presents a generalized channel probing (GCP) method and a generalized pre-processing (GPP) method as two consecutive frontend steps for secret key generation from a MIMO channel between two multi-antenna legitimate nodes against a multi-antenna eavesdropper (Eve). The degree of freedom (DoF) of the secret key capacity (SKC) of GCP/GPP are highlighted and discussed. If the number of antennas on Eve is larger than or equal to the larger number of the antennas on the two legitimate nodes, the SKC-DoF of GCP/GPP within each coherence period equals its minimum, which is either zero for non-reciprocal channel or the product of the numbers of antennas on the two legitimate nodes for reciprocal channel. Otherwise, the SKC-DoF of GCP/GPP increases with the number of random transmissions in GCP within each coherence period regardless of the channel reciprocity. A computational algorithm required for GPP is also discussed, and its performance illustrated via simulations.
MIMO-OFDM as well as MIMO systems are two steering technologies for 5G and beyond communication systems. The key problem with conventional method is that the security protocols used does not take any advantage of the wireless channel features, making eavesdropping a serious issue. This poses security issues at the physical layer. In this paper, we use SecretKey Capacity (SKC) on MIMO and MIMO-OFDM systems with more than one receiver antennas for the eavesdropper. We investigate the impact of channel estimation errors on the SKC performance. We derive the closed form expressions for SKC for both the MIMO and MIMO-OFDM channel The effect of channel estimation errors at the legitimate receiver end is also investigated for different MIMO-OFDM configurations. MIMO-OFDM channels can achieve positive SKC even when eavesdropper has higher number of receiver antenna than legitimate receiver, which is not possible in the case of MIMO channels. Increase in the channel estimation errors can increase SKC of the system. SKC results are discussed here with different number of transmit and receive antenna for eavesdropper and legitimate receiver.
This chapter considers securing wireless communications at the physical layer using multiple antennas. In particular, the multiple antenna (MIMO) wiretap channel is presented and its secrecy capacity is provided. Further investigation of the high-SNR characterization of the secrecy capacity, i.e., secure degrees of freedom, reveals that the secrecy capacity of the MIMO wiretap channel does not scale with the transmit power when the eavesdropper has an advantage over the legitimate transmitter in the number of antennas. An external multi-antenna cooperative jammer terminal is introduced to the channel model in order to improve its secrecy capacity scaling with power. The secure degrees of freedom for this new multi-terminal multi-antenna channel is characterized. While for these models the eavesdropper channel state information is known at the legitimate terminals, the chapter next removes this assumption and presents results that utilize multiple antennas in order to provide secure communications irrespective of the eavesdropper channel state. An achievable strong secrecy rate and the secure degrees of freedom characterization for the MIMO wiretap channel when the eavesdropper channel is arbitrary, varying, and unknown at the legitimate terminals are presented. Finally, the extension of this model to the two-user multiple access MIMO wiretap channel is considered and its secure degrees of freedom region is characterized. Introduction The wiretap channel and the notion of secrecy capacity, introduced in [1] and generalized by [2], have provided the framework for physical layer design for wireless channels with information theoretic guarantees against eavesdropping. A natural model in wireless communications is one which utilizes multiple antennas for transmission and reception at each node. The improvement in secrecy rate that can be obtained using multiple antennas in a point-to-point channel with an eavesdropper has been studied extensively in several references, including [3, 4]. Specifically, the secrecy capacity of the Gaussian model consisting of a multiple antenna legitimate transmitter, a multiple antenna legitimate receiver, and a multiple antenna eavesdropper, termed the multiple-input multiple-output (MIMO) wiretap channel, has been identified in full generality in [5, 6].
In this paper, we consider a massive MIMO wiretap system where the transmitter, the receiver and the eavesdropper are equipped with a large number of antennas. Being different from the previous works using asymptotic random matrix theory, our analysis relies on the concentration measure of non-asymptotic random matrix theory which allows us to obtain tight bounds for secrecy capacity of massive MIMO system with finite antenna number. The analytical and simulation results reveal the following, in the massive MIMO system employing equal power allocation at each transmit antenna: 1) the secrecy capacity falls within a bounds with a probability growing exponentially with the number of transmit antenna, while the ergodic secrecy capacity falls within a deterministic bounds; 2) the gap between the upper and the lower bound on secrecy rate is proportional to the square root of the SNR at legitimate receiver and the SNR at eavesdropper, respectively; 3) when the entry of legitimate channel matrix and eavesdropping channel matrix satisfies Gaussian distribution, the gap between the upper and the lower bound on secrecy rate is a linear reciprocal function of the number of transmit antennas.
In this paper, we consider secure transmissions in ergodic Rayleigh fast-faded multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channels with only statistical channel state information at the transmitter (CSIT). When the antennas at the legitimate receiver are more than (or equal to) those at the eavesdropper, we prove the first MIMOME secrecy capacity with partial CSIT by establishing a new secrecy capacity upper-bound. The key step is to form an MIMOME degraded channel by dividing the legitimate receiver's channel matrix into two submatrices, and setting one of the submatrices to be the same as the eavesdropper's channel matrix. Next, under the total power constraint over all transmit antennas, we analytically solve the channel-input covariance matrix optimization problem to fully characterize the MIMOME secrecy capacity. Typically, the MIMOME optimization problems are non-concave. However, thanks to the proposed degraded channel, we can transform the stochastic MIMOME optimization problem to be a Schur-concave one and then find its solution. Besides total power constraint, we also investigate the secrecy capacity when the transmitter is subject to the practical per-antenna power constraint. The corresponding optimization problem is even more difficult since it is not Schur-concave. Under the two power constraints considered, the corresponding MIMOME secrecy capacities can both scale with the signal-to-noise ratios (SNR) when the difference between numbers of antennas at legitimate receiver and eavesdropper are large enough. However, when the legitimate receiver and eavesdropper have a single antenna each, such SNR scalings do not exist for both cases.