Cryptographic puzzles are moderately difficult problems that can be solved by investing non-trivial amounts of computation and/or storage. Devising models for cryptographic puzzles has only recently started to receive attention from the cryptographic community as a first step towards rigorous models and proofs of security of applications that employ them (e.g. Denial-of-service (DoS) resistance). Unfortunately, the subtle interaction between the complex scenarios for which cryptographic puzzles are intended and typical difficulties associated with defying concrete security easily leads to flaws in definitions and proofs. Indeed, as a first contribution we exhibit shortcomings of the state-of-the-art definition of security of cryptographic puzzles and point out some flaws in existing security proofs. The main contribution of this paper are new security definitions for puzzle difficulty. We distinguish and formalize two distinct flavors of puzzle security (which we call optimal and ideal) and in addition properly define the relation between solving one puzzle vs. solving multiple ones. We demonstrate the applicability of our notions by analyzing the security of two popular puzzle constructions. In addition, we briefly investigate existing definitions for the related notion of DoS security. We demonstrate that the only rigorous security notions proposed to date is not sufficiently demanding (as it allows to prove secure protocols that are clearly not DoS resilient) and suggest an alternative definition. Our results are not only of theoretical interest. We show that our better characterization of hardness for puzzles and DoS resilience allows establishing formal bounds on the effectiveness of client puzzles which confirm previous empirical observations.
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