  The problem of secure communication in Amplify-and-Forward (AF) relay
networks with multiple eavesdroppers is considered. Assuming that a receiver
(destination or eavesdropper) can decode a message only if the received SNR is
above a predefined threshold, we introduce SNR based optimization formulations
to calculate optimal scaling factors for relay nodes in two scenarios. In the
first scenario, we maximize the achievable rate at the legitimate destination,
subject to the condition that the received SNR at each eavesdropper is below
the target threshold. Due to the non-convex nature of the objective function
and eavesdroppers' constraints, we transform variables and obtain a
Quadratically Constrained Quadratic Program (QCQP) with convex constraints,
which can be solved efficiently. When the constraints are not convex, we
consider a Semi-definite relaxation (SDR). In the second scenario, we minimize
the total power consumed by all relay nodes, subject to the condition that the
received SNR at the legitimate destination is above the threshold and at every
eavesdropper, it is below the corresponding threshold. We propose a
semi-definite relaxation of the problem in this scenario and also provide an
analytical lower bound.
