Approximating Cumulative Pebbling Cost is Unique Games Hard
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The cumulative pebbling complexity of a directed acyclic graph G is defined as cc(G) = _P ∑_i |P_i|, where the minimum is taken over all legal (parallel) black pebblings of G and |P_i| denotes the number of pebbles on the graph during round i. Intuitively, cc(G) captures the amortized Space-Time complexity of pebbling m copies of G in parallel. The cumulative pebbling complexity of a graph G is of particular interest in the field of cryptography as cc(G) is tightly related to the amortized Area-Time complexity of the data-independent memory hard function (iMHF) f_G,H [AS15] defined using a constant indegree directed acyclic graph (DAG) G and a random oracle H(·). A secure iMHF should have amortized Space-Time complexity as high as possible e.g., to deter brute-force password attacker who wants to find x such that f_G,H(x) = h. Thus, to analyze the (in)security of a candidate iMHF f_G,H, it is crucial to estimate the value cc(G) but currently, upper and lower bounds for leading iMHF candidates differ by several orders of magnitude. Blocki and Zhou recently showed that is NP-Hard to compute cc(G), but their techniques do not even rule out an efficient (1+ϵ)-approximation algorithm for any constant ϵ>0. We show that for any constant c>0, it is Unique Games hard to approximate cc(G) to within a factor of c. (See the paper for the full abstract.)
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