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Although the virtual collocation of service providers (SPs) on top of infrastructure providers (InPs) via network virtualization brings various benefits, we posit that operational confidentiality has not been considered in this network model. We extend and apply the Secure Multiparty Computation (SMC) protocol to solving Minimum Disclosure Routing (MDR), namely, enabling an SP to route packets without disclosing routing information to InPs. We implement the proposed MDR protocol and evaluate its performance via experiments by comparing it against the prediction based on our analytical performance model. Our study reveals that MDR can be securely achieved with marginal latency overhead with regard to the convergence time in well-engineered nonsecure routing algorithms. Our study sheds light on the path for network virtualization to be used to resolve the challenges for the ISPs of today.