Secure cascade channel synthesis

We investigate channel synthesis in a cascade setting where nature provides an iid sequence Xⁿ at node 1. Node 1 can send a message at rate R₁ to node 2 and node 2 can send a message at rate R₂ to node 3. Additionally, all 3 nodes share bits of common randomness at rate R₀. We want to generate sequences Y ⁿ and Zⁿ along nodes in the cascade such that (Xⁿ, Y ⁿ, Zⁿ) appears to be appropriately correlated and iid even to an eavesdropper who is cognizant of the messages being sent. We characterize the optimal tradeoff between the amount of common randomness used and the required rates of communication. We also solve the problem for arbitrarily long cascades and provide an inner bound for cascade channel synthesis without an eavesdropper.