Proofs of Work From Worst-Case Assumptions
We give Proofs of Work (PoWs) whose hardness is based on well-studied worst-case assumptions from fine-grained complexity theory. This extends the work of (Ball et al., STOC ’17), that presents PoWs that are based on the Orthogonal Vectors, 3SUM, and All-Pairs Shortest Path problems. These, however, were presented as a ‘proof of concept’ of provably secure PoWs and did not fully meet the requirements of a conventional PoW: namely, it was not shown that multiple proofs could not be generated faster than generating each individually. We use the considerable algebraic structure of these PoWs to prove that this non-amortizability of multiple proofs does in fact hold and further show that the PoWs’ structure can be exploited in ways previous heuristic PoWs could not.
This creates full PoWs that are provably hard from worst-case assumptions (previously, PoWs were either only based on heuristic assumptions or on much stronger cryptographic assumptions (Bitansky et al., ITCS ’16)) while still retaining significant structure to enable extra properties of our PoWs. Namely, we show that the PoWs of (Ball et al., STOC ’17) can be modified to have much faster verification time, can be proved in zero knowledge, and more.
Finally, as our PoWs are based on evaluating low-degree polynomials originating from average-case fine-grained complexity, we prove an average-case direct sum theorem for the problem of evaluating these polynomials, which may be of independent interest. For our context, this implies the required non-amortizability of our PoWs.
We are grateful to Oded Goldreich and Guy Rothblum for clarifying definitions of direct sum theorems, and for the suggestion of using interaction to increase the gap between solution and verification in our PoWs. We would also like to thank Tal Moran and Vinod Vaikuntanathan for several useful discussions. We also thank the anonymous reviewers for comments and references.
The bulk of this work was performed while the authors were at IDC Herzliya’s FACT center and supported by NSF-BSF Cyber Security and Privacy grant #2014/632, ISF grant #1255/12, and by the ERC under the EU’s Seventh Framework Programme (FP/2007-2013) ERC Grant Agreement #07952. Marshall Ball is supported in part by the Defense Advanced Research Project Agency (DARPA) and Army Research Office (ARO) under Contract #W911NF-15-C-0236, NSF grants #CNS-1445424 and #CCF-1423306, the Leona M. & Harry B. Helmsley Charitable Trust, ISF grant no. 1790/13, and the Check Point Institute for Information Security. Alon Rosen is also supported by ISF grant no. 1399/17. Manuel Sabin is also supported by the National Science Foundation Graduate Research Fellowship under Grant #DGE-1106400. Prashant Nalini Vasudevan is also supported by the IBM Thomas J. Watson Research Center (Agreement #4915012803), by NSF Grants CNS-1350619 and CNS-1414119, and by the Defense Advanced Research Projects Agency (DARPA) and the U.S. Army Research Office under contracts W911NF-15-C-0226 and W911NF-15-C-0236.
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