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We study power allocation for secrecy rate maximization in a multicarrier decode-and-forward relay system, where an eavesdropper exists. We consider three transmission modes: in no communication, the source and the relay do not transmit at all; in direct communication, the source broadcasts signals during the first time slot and the relay does not forward any signal during the second time slot; in relay communication, the source broadcasts signals during the first time slot and the relay forwards the reencoded signals to the destination during the second time slot. Determining the transmission strategy adaptively on each subcarrier, the optimal source power and the optimal relay power over all subcarriers are derived to maximize the sum secrecy rate under a total system power constraint. In addition, a suboptimal power allocation scheme is proposed to substantially reduce the computational complexity. It is shown that the proposed suboptimal solution is asymptotically optimal in the limit as the number of subcarriers goes to infinity. Extensive numerical results are presented for various scenarios. In particular, the performance of the suboptimal scheme is very close to that of the optimal scheme even if the number of subcarriers is moderately small.