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

Beam-forming for Secure Communication in Amplify-and-Forward Networks: An SNR based approach