  Unicast communication over a network of $M$-parallel relays in the presence
of an eavesdropper is considered. The relay nodes, operating under individual
power constraints, amplify and forward the signals received at their inputs.
The problem of the maximum secrecy rate achievable with AF relaying is
addressed. Previous work on this problem provides iterative algorithms based on
semidefinite relaxation. However, those algorithms result in suboptimal
performance without any performance and convergence guarantees. We address this
problem for three specific network models, with real-valued channel gains. We
propose a novel transformation that leads to convex optimization problems. Our
analysis leads to (i)a polynomial-time algorithm to compute the optimal secure
AF rate for two of the models and (ii) a closed-form expression for the optimal
secure rate for the other.
