Secure Degrees of Freedom of MIMO Two-Way Wiretap Channel With no CSI Anywhere

This article considers a two-way multiple-input multiple-output (MIMO) Rayleigh block fading wiretap channel with two full-duplex nodes (Alice and Bob) exchanging messages simultaneously and wiretapped by a <inline-formula> <tex-math notation="LaTeX">$ {N}_{ {e}}$ </tex-math></inline-formula>-antennas eavesdropper (Eve). The channel state remains unchanged over a coherence interval T and is unknown to all terminals, including Alice, Bob, and Eve, at the beginning of the coherence interval. We first show the expression of the optimal signal waveform for Alice and Bob, which maximizes the sum secrecy rate, should be the products of independent random diagonal matrices and isotropically distributed unitary matrices. Then, conditioned on large T and small <inline-formula> <tex-math notation="LaTeX">$ {N}_{ {e}}$ </tex-math></inline-formula>, the upper-bound for the secure degree of freedom (s.d.o.f.) of the two-way wiretap channel is derived, and a constant-norm-signaling (CNS) scheme is presented to achieve this theoretical bound. Finally, we formulate an optimization problem of the s.d.o.f. achieved by the CNS scheme for general cases with arbitrary T and <inline-formula> <tex-math notation="LaTeX">$ {N}_{ {e}}$ </tex-math></inline-formula>, and solve this problem by exploring the monotonicity of the achievable s.d.o.f.. Our result shows that the optimal s.d.o.f. can be independent of coherence time T and becomes the product of the numbers of legitimate nodes’s transmitter antennas (NLNTA) for large <inline-formula> <tex-math notation="LaTeX">$ {N}_{ {e}}$ </tex-math></inline-formula> and small NLNTA.