  Authenticated data structures provide cryptographic proofs that their answers
are as accurate as the author intended, even if the data structure is being
controlled by a remote untrusted host. We present efficient techniques for
authenticating data structures that represent graphs and collections of
geometric objects. We introduce the path hash accumulator, a new primitive
based on cryptographic hashing for efficiently authenticating various
properties of structured data represented as paths, including any decomposable
query over sequences of elements. We show how to employ our primitive to
authenticate queries about properties of paths in graphs and search queries on
multi-catalogs. This allows the design of new, efficient authenticated data
structures for fundamental problems on networks, such as path and connectivity
queries over graphs, and complex queries on two-dimensional geometric objects,
such as intersection and containment queries.
