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We develop new theoretical tools for proving lower-bounds on the (amortized) complexity of functions in a parallel setting. We demonstrate their use by constructing the first provably secure Memory-hard functions (MHF); a class of functions recently gaining acceptance in practice as an effective means to counter brute-force attacks on security relevant functions. The work was done primary while both authors were at ETH Zürich.doi:10.1145/2746539.2746622 dblp:conf/stoc/AlwenS15 fatcat:mlcyoknonveyvcq46q2icczwjq