Parallel Quantum Pebbling: Analyzing the Post-Quantum Security of iMHFs
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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 𝒳, i.e., given y ∈{0,1}^* find x ∈𝒳 such that f_G,H(x)=y. While a classical attacker will need to evaluate the function f_G,H at least m=|𝒳| times a quantum attacker running Grover's algorithm only requires 𝒪(√(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) quantum pebbling game which captures additional restrictions imposed by the No-Deletion Theorem in Quantum Computing. We apply our new quantum pebbling game to analyze the quantum space-time complexity of several important graphs: the line graph, Argon2i-A, Argon2i-B, and DRSample. (See the paper for the full abstract.)
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