On the Memory-Hardness of Data-Independent Password-Hashing Functions

We show attacks on five data-independent memoryhard functions (iMHF) that were submitted to the password hashing competition. Informally, an MHF is a function which cannot be evaluated on dedicated hardware, like ASICs, at significantly lower energy and/or hardware cost than evaluating a single instance on a standard single-core architecture. Data-independent means the memory access pattern of the function is independent of the input; this makes iMHFs harder to construct than data-dependent ones, but the latter can be attacked by various side-channel attacks. Following [Alwen-Blocki’16], we capture the evaluation of an iMHF as a directed acyclic graph (DAG). The cumulative parallel pebbling complexity of this DAG is a good measure for the cost of evaluating the iMHF on an ASIC. If n denotes the number of nodes of a DAG (or equivalently, the number of operations — typically hash function calls — of the underlying iMHF), its pebbling complexity must be close to n for the iMHF to be memory-hard. We show that the following iMHFs are far from this bound: Rig.v2, TwoCats and Gambit can be attacked with complexity O(n); the data-independent phase of Pomelo (a finalist of the password hashing competition) and Lyra2 (also a finalist) can be attacked with complexity O(n) and O(n), respectively. For our attacks we use and extend the technique developed by [Alwen-Blocki’16], who show that the pebbling complexity of a DAG can be upper bounded in terms of its depthrobustness.