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 \mathcalX, i.e., given y ∈{0,1}^* find x ∈\mathcalX such that f_G,H(x)=y. While a classical attacker will need to evaluate the function f_G,H at least m=|\mathcalX| times a quantum attacker running Grover’s algorithm only requires Ø\sqrtm 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 for 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. Specifically, (1) we show that a line graph of size N has reversible space-time complexity at most ØN^1+\frac2\sqrt\log N. (2) We show that any (e,d)-reducible DAG has reversible space-time complexity at most ØNe+dN2^d. In particular, this implies that the reversible space-time complexity of Argon2i-A and Argon2i-B are at most ØN^2 \log \log N/\sqrt\log N and ØN^2/\sqrt[3]\log N, respectively. (3) We show that the reversible space-time complexity of DRSample is at most ØN^2 \log \log N/\log N. We also study the cumulative pebbling cost of reversible pebblings extending a (non-reversible) pebbling attack of Alwen and Blocki on depth-reducible graphs.