arXiv subjects

Blake Holman

Publications and source records attributed to Blake Holman.

3 recordsLinked to original sources

arXiv2025

Data-Dependent Memory-Hard Functions: Sustained Space and Cumulative Complexity Trade-offs in the Parallel Random Oracle Model

Memory-Hard Functions (MHFs) protect passwords and other low-entropy secrets against brute-force attacks. Sustained space complexity (SSC), the strongest natural formalization of memory hardness, measures how long an attacker's memory remains above a threshold. Since no function computable in sequential time $\Theta(N)$ can force every parallel attacker to sustain $\Theta(N)$ memory for $\Theta(N)$ steps, the appropriate goal is a strong tradeoff between SSC and cumulative memory complexity (CMC). Blocki and Holman (CRYPTO 2022) established such tradeoffs in the dynamic pebbling model, but their construction used expensive combinatorial graphs, and the pebbling abstraction does not rule out more efficient attacks in the stronger Parallel Random Oracle Model (PROM). We address both limitations. We construct a data-dependent MHF, DEGSample, and prove the first SSC/CMC tradeoff for data-dependent MHFs directly in the PROM. In the dynamic pebbling model, every strategy either sustains $\Omega(N)$ memory for $\Omega(N)$ steps or incurs the maximal CMC penalty $\Omega(N^{3-\epsilon})$. In the PROM, every attacker either sustains $\Omega(N)$ memory for $\Omega(N)$ steps or incurs CMC at least $\Omega(N^{2.5-\epsilon})$. We introduce ancestral robustness and show that, together with fractional depth-robustness, it yields strong PROM tradeoffs under a natural dynamization procedure. The lower bound combines a time-space tradeoff with an extraction procedure converting any PROM execution into a cost-equivalent pebbling of the realized graph.

cs.CR↗

arXiv2025

Quantum Search with In-Place Queries

Quantum query complexity is typically characterized in terms of XOR queries |x,y> to |x,y+f(x)> or phase queries, which ensure that even queries to non-invertible functions are unitary. When querying a permutation, another natural model is unitary: in-place queries |x> to |f(x)>. Some problems are known to require exponentially fewer in-place queries than XOR queries, but no separation has been shown in the opposite direction. A candidate for such a separation was the problem of inverting a permutation over N elements. This task, equivalent to unstructured search in the context of permutations, is solvable with $O(\sqrt{N})$ XOR queries but was conjectured to require $\Omega(N)$ in-place queries. We refute this conjecture by designing a quantum algorithm for Permutation Inversion using $O(\sqrt{N})$ in-place queries. Our algorithm achieves the same speedup as Grover's algorithm despite the inability to efficiently uncompute queries or perform straightforward oracle-controlled reflections. Nonetheless, we show that there are indeed problems which require fewer XOR queries than in-place queries. We introduce a subspace-conversion problem called Function Erasure that requires 1 XOR query and $\Theta(\sqrt{N})$ in-place queries. Then, we build on a recent extension of the quantum adversary method to characterize exact conditions for a decision problem to exhibit such a separation, and we propose a candidate problem.

quant-ph↗

arXiv2021

The Parallel Reversible Pebbling Game: Analyzing the Post-Quantum Security of iMHFs

The classical (parallel) black pebbling game is a useful abstraction which allows us to analyze the resources (space, space-time, cumulative space) necessary to evaluate a function $f$ with a static data-dependency graph $G$. Of particular interest in the field of cryptography are data-independent memory-hard functions $f_{G,H}$ which are defined by a directed acyclic graph (DAG) $G$ and a cryptographic hash function $H$. The pebbling complexity of the graph $G$ characterizes the amortized cost of evaluating $f_{G,H}$ multiple times as well as the total cost to run a brute-force preimage attack over a fixed domain $\mathcal{X}$, i.e., given $y \in \{0,1\}^*$ find $x \in \mathcal{X}$ such that $f_{G,H}(x)=y$. While a classical attacker will need to evaluate the function $f_{G,H}$ at least $m=|\mathcal{X}|$ times a quantum attacker running Grover's algorithm only requires $\mathcal{O}(\sqrt{m})$ blackbox calls to a quantum circuit $C_{G,H}$ evaluating the function $f_{G,H}$. Thus, to analyze the cost of a quantum attack it is crucial to understand the space-time cost (equivalently width times depth) of the quantum circuit $C_{G,H}$. We first observe that a legal black pebbling strategy for the graph $G$ does not necessarily imply the existence of a quantum circuit with comparable complexity -- in contrast to the classical setting where any efficient pebbling strategy for $G$ corresponds to an algorithm with comparable complexity evaluating $f_{G,H}$. Motivated by this observation we introduce a new parallel reversible pebbling game which captures additional restrictions imposed by the No-Deletion Theorem in Quantum Computing. We apply our new reversible pebbling game to analyze the reversible space-time complexity of several important graphs: Line Graphs, Argon2i-A, Argon2i-B, and DRSample. (See the paper for the full abstract.)

quant-ph↗