Bound on the Key Establishment Rate for Multi-Antenna Reciprocal Electromagnetic Channels

Recent research on key establishment using estimates of the reciprocal electromagnetic channel provides an optimal training approach that maximizes the key rate achievable when each node uses a beamformed antenna array. The work detailed here generalizes this concept by allowing the antennas at the two nodes to be represented by arbitrary transmit current or receive weight distributions modeled using complete basis sets and constrained to reside within fixed apertures. The representation requires that we modify the key rate optimization formulation to incorporate mutual coupling and spatially correlated estimation errors. Application of this modified optimization technique produces both the coefficients of the basis functions representing the current distributions and the optimal training approach that together maximize the achievable key rate. Simulations are used to demonstrate the convergence of the technique as the number of basis functions grows large as well as the behavior of the key rate upper bound for different scenarios.