Maximization of Worst-Case Secret Key Rates in MIMO Systems with Eavesdropper Anne Wolf and Eduard A. Jorswieck Communications Theory, Communications Laboratory Department of Electrical Engineering and Information Technology Dresden University of Technology, Germany Email: {anne.wolf, eduard.jorswieck}@tu-dresden.de Abstract—A multi-antenna (MIMO) system with eavesdropper is studied, where the transmitter has no perfect knowledge about the channel to the eavesdropper. The transmitter only knows that the channel matrix of the eavesdropper is drawn from a certain set. For this scenario, the worst-case secret key rate and the worst-case channel matrix of the eavesdropper are characterized. It is shown that the maximization of this rate under a sum power constraint at the transmitter is a saddle point problem. Upper and lower bounds for the maximized worst-case secret key rate are derived. Strategies for the high and low SNR regime are presented and the results are finally illustrated and discussed in comparison with the maximized worst-case secrecy rate. Index Terms—worst-case secret key rate, secret key rate max- imization, MIMO system with eavesdropper, wiretap channel. I. I NTRODUCTION A. Scenario and problem statement We consider the problem of secret-key agreement via a noisy and wiretapped communication channel. The transmitter Alice and an intended receiver Bob want to agree on a common random key, which should be kept perfectly secret from an eavesdropper Eve. Alice generates a sequence X n , which consists of n symbols, and transmits this sequence over a wireless channel to the intended receiver Bob, who receives the sequence Y n . This transmission is eavesdropped by a second receiver, the eavesdropper Eve, who receives the sequence Z n . Additionally, Alice and Bob can communicate over a public channel. These transmissions are completely known to the eavesdropper. Finally, Alice and Bob calculate random keys based on all the information that is available to them, and these two keys should be equal with high probability. We study a MIMO scenario, where the transmitter aims to maximize the worst-case secret key rate that is achievable under a sum power constraint and for a given set of possible eavesdropper channels. This set of channels models non-line- of-sight (NLOS) channels, where the link quality depends on effects like multi-path propagation rather than on the distance of the antennas and the free-space loss. B. Secret key rate and capacity There are four basic models for the secret-key agreement between two users. An introduction to the concepts of secret- key agreement and an overview of the basic models together This work is supported in part by the German Research Foundation (DFG) under grant JO 801/2-2. with the results on achievable secret key rates and the secret key capacity can be found in [1]. In this paper, we focus on the wiretapped channel-type model, i.e., the CW model, which consists of a noisy wiretap channel and a noiseless public channel. The CW model with discrete memoryless channels was introduced in [2]. In such a system, the secret key capacity C K is upper bounded by C K ≤ max pX I (X; Y |Z ), (1) where p X is the probability mass function of the random variable X. The bound is tight if the random variables X, Y , and Z form a Markov chain in any order. If the random variables X, Y , and Z form a Markov chain in the order Y → X → Z , the secret key capacity equals the backward secret key capacity, which is the largest key rate possible if only one backward transmission, i.e., a transmission from the intended receiver to the transmitter over the public channel, is permitted, and is given by [2] C K = max pX ( I (X; Y ) − I (Y ; Z ) ) . (2) In [3], the CW model was extended to the case with continuous alphabets and average power constraint. In such a system, the secret key capacity C K is given by C K = max pX with E(|X| 2 )≤P ( I (X; Y ) − I (Y ; Z ) ) (3) where p X is the probability density function of the random variable X. C. Notation Throughout the paper, we use the following notation: The expected value of the random variable X is denoted by E(X). The set of all positive real numbers is R + = {x ∈ R | x> 0}. The identity matrix of dimension n × n is I n . The Hermi- tian transpose matrix of the matrix A is A H . The notation A  0 means that the matrix A is positive-semidefinite and consequently Hermitian. The indicator function is denoted by I A (x), which has the value 1 for all x ∈A and the value 0 for all x/ ∈A. We define [x] + = max{0,x} = x I R+ (x), and we introduce x =(x k ) n k=1 and A =(a ij ) n i,j=1 as notation for the vector x =(x 1 ,x 2 ,...,x n ) of length n and the square matrix A of dimension n × n, respectively.