A Physical-Layer Key Generation Scheme for Non-Square Channel Matrices Based on PMI

Physical-layer secret key generation (PKG) is an emerging technique to address the secret key distribution problem and precoding matrix index (PMI)-based PKG approaches can thwart eavesdropping attempts. However, existing PKG schemes based on PMI only considered square channel matrices, which restricts the practical application. In this paper, we propose a new PKG scheme called NS-PMI-PKG which can yield secret keys for non-square channel matrices in time-division duplex (TDD) mode. Specifically, NS-PMI-PKG includes two vital steps, i.e., codebook generation and detection. First, the codebook generation approach ensures a unitary precoding matrix by applying singular value decomposition (SVD) to the precoding matrix. Second, we present an efficient codebook detection approach leveraging the properties of unitary matrices and the concept of template matching. Subsequently, we present a theoretical analysis of the proposed NS-PMI-PKG scheme from the aspects of key generation correctness, security, and efficiency. The simulation results demonstrate that the proposed scheme exhibits good key generation performance when the channel matrix is non-square. At the signal-to-noise ratio (SNR) of 25 dB, the bit error rate (BER) of the proposed scheme is approximately an order of magnitude lower than that of the baseline scheme. In addition, the eavesdropper struggles to obtain any keys when 2 meters away from the legitimate user, and even when sufficiently close, can acquire at most half the keys. Also, the generated key exhibits good randomness, enhancing its resistance to brute force cracking.