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In this work, we have examined the distributed consensus averaging problem from a novel point of view considering the need for privacy and anonymity. We have proposed a method for incorporating security into the scalable average consensus mechanisms proposed in the literature. Random Offsets Method (ROM) is lightweight, transparent and flexible since it is not based on cryptography, does not require any change in the fusion system and can be used optionally by some nodes who care about their privacy. In this method, which is based on noisification of nodes' information, we achieve robustness against n - 1 colluding adversaries in a network of in nodes, which is maximum level of robustness against collusions. Convergence and collusion robustness of ROM are analyzed mathematically and through simulation.