# Cryptographic puzzles and DoS resilience, revisited

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## Abstract

Cryptographic puzzles (or client 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 toward 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 defining 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 optimality and fairness and in addition, properly define the relation between solving one puzzle versus solving multiple ones. We demonstrate the applicability of our notions by analyzing the security of two popular puzzle constructions. We briefly investigate existing definitions for the related notion of security against DoS attacks. We demonstrate that the only rigorous security notion proposed to date is not sufficiently demanding (as it allows to prove secure protocols that are clearly not DoS resistant) and suggest an alternative definition. Our results are not only of theoretical interest: the better characterization of hardness for puzzles and DoS resilience allows establishing formal bounds on the effectiveness of client puzzles which confirm previous empirical observations. We also underline clear practical limitations for the effectiveness of puzzles against DoS attacks by providing simple rules of thumb that can be easily used to discard puzzles as a valid countermeasure for certain scenarios.

## Keywords

Client puzzle DoS resilience PoW protocols## Mathematics Subject Classification (2010)

94A60## Notes

### Acknowledgments

We thank Douglas Stebila and to the anonymous referees for their comments and feedback on our work. First author was partially supported by National Research Grants CNCSIS UEFISCDI, Project Number PNII IDEI 940/2008–2011 and POSDRU/21/1.5/G/13798, inside POSDRU Romania 2007–2013.

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