If it is deemed acceptable for per-district tallies to be made public, running a hierarchical election becomes a simple matter of running several parallel district-wide elections. Any existing election protocol with the desired properties can be used for the district-wide elections, resulting in public district tallies that can then be added up by the general public. No global authorities are needed in this scheme.
If sum secret sharing (as opposed to Shamir secret sharing) is used, and it is desired to further reduce communication complexity in this scheme, a tradeoff reducing both complexity and potential privacy can be made as follows. Instead of every voter contacting all Tintermediate authorities in the district, the voter can contact only S < T authorities. As before, the voter still engages in an interactive proof to show that the ballot is valid; however, for all intermediate authorities not explicitly contacted by the voter, the component type for that authority defaults to 0 mod r (i.e., the encrypted component defaults to 1 mod nj for authority tj). While obviously reducing the amount of communication in the protocol, this change also reduces the maximum number of corrupt authorities needed to violate a voter's privacy from T to S. Depending on whether S is a large number itself, this may or may not be a sensible tradeoff to make. We term this tradeoff subset authority voting. Unfortunately, subset authority voting only works for sum secret sharing, not Shamir secret sharing.