International Association
for Cryptologic Research

We study the space complexity of the two related fields of {\em differential privacy} and {\em adaptive data analysis}. Specifically, \begin{enumerate} \item Under standard cryptographic assumptions, we show that there exists a problem $P$ that requires exponentially more space to be solved efficiently with differential privacy, compared to the space needed without privacy. To the best of our knowledge, this is the first separation between the space complexity of private and non-private algorithms. \item The line of work on adaptive data analysis focuses on understanding the number of {\em samples} needed for answering a sequence of adaptive queries. We revisit previous lower bounds at a foundational level, and show that they are a consequence of a space bottleneck rather than a sampling bottleneck. \end{enumerate} To obtain our results, we define and construct an encryption scheme with multiple keys that is built to withstand a limited amount of key leakage in a very particular way.

Memory-hard functions (MHFs) are a key cryptographic primitive underlying the design of moderately expensive password hashing algorithms and egalitarian proofs of work. Over the past few years several increasingly stringent goals for an MHF have been proposed including the requirement that the MHF have high sequential space-time (ST) complexity, parallel space-time complexity, amortized area-time (aAT) complexity and sustained space complexity. Data-Independent Memory Hard Functions (iMHFs) are of special interest in the context of password hashing as they naturally resist side-channel attacks. iMHFs can be specified using a directed acyclic graph (DAG) G with $$N=2^n$$ nodes and low indegree and the complexity of the iMHF can be analyzed using a pebbling game. Recently, Alwen et al. [ABH17] constructed a DAG called DRSample that has aAT complexity at least $$\varOmega \!\left( N^2/{\text {log}} N\right) $$ . Asymptotically DRSample outperformed all prior iMHF constructions including Argon2i, winner of the password hashing competition (aAT cost $${\mathcal {O}} \!\left( N^{1.767}\right) $$ ), though the constants in these bounds are poorly understood. We show that the greedy pebbling strategy of Boneh et al. [BCS16] is particularly effective against DRSample e.g., the aAT cost is $${\mathcal {O}} (N^2/{\text {log}} N)$$ . In fact, our empirical analysis reverses the prior conclusion of Alwen et al. that DRSample provides stronger resistance to known pebbling attacks for practical values of $$N \le 2^{24}$$ . We construct a new iMHF candidate (DRSample+BRG) by using the bit-reversal graph to extend DRSample. We then prove that the construction is asymptotically optimal under every MHF criteria, and we empirically demonstrate that our iMHF provides the best resistance to known pebbling attacks. For example, we show that any parallel pebbling attack either has aAT cost $$\omega (N^2)$$ or requires at least $$\varOmega (N)$$ steps with $$\varOmega (N/{\text {log}} N)$$ pebbles on the DAG. This makes our construction the first practical iMHF with a strong sustained space-complexity guarantee and immediately implies that any parallel pebbling has aAT complexity $$\varOmega (N^2/{\text {log}} N)$$ . We also prove that any sequential pebbling (including the greedy pebbling attack) has aAT cost $$\varOmega \!\left( N^2\right) $$ and, if a plausible conjecture holds, any parallel pebbling has aAT cost $$\varOmega (N^2 \log \log N/{\text {log}} N)$$ —the best possible bound for an iMHF. We implement our new iMHF and demonstrate that it is just as fast as Argon2. Along the way we propose a simple modification to the Argon2 round function that increases an attacker’s aAT cost by nearly an order of magnitude without increasing running time on a CPU. Finally, we give a pebbling reduction that proves that in the parallel random oracle model (PROM) the cost of evaluating an iMHF like Argon2i or DRSample+BRG is given by the pebbling cost of the underlying DAG. Prior pebbling reductions assumed that the iMHF round function concatenates input labels before hashing and did not apply to practical iMHFs such as Argon2i, DRSample or DRSample+BRG where input labels are instead XORed together.