Collapseability of Tree Hashes

One oft-endeavored security property for cryptographic hash functions is collision resistance: it should be computationally infeasible to find distinct inputs \(x,x'\) such that \(H(x) = H(x')\), where H is the hash function. Unruh (EUROCRYPT 2016) proposed collapseability as its quantum equivalent. The Merkle-Damgård and sponge hashing modes have recently been proven to be collapseable under the assumption that the underlying primitive is collapseable. These modes are inherently sequential. In this work, we investigate collapseability of tree hashing. We first consider fixed length tree hashing modes, and derive conditions under which their collapseability can be reduced to the collapseability of the underlying compression function. Then, we extend the result to two methods for achieving variable length hashing: tree hashing with domain separation between message and chaining value, and tree hashing with length encoding at the end of the tree. The proofs are performed using the collapseability composability framework of Fehr (TCC 2018), that allows us to discard of deeply technical quantum details and to focus on proper composition of the tree hashes from their compression function.