# Revisiting the Cryptographic Hardness of Finding a Nash Equilibrium

@inproceedings{Garg2016RevisitingTC, title={Revisiting the Cryptographic Hardness of Finding a Nash Equilibrium}, author={Sanjam Garg and Omkant Pandey and Akshayaram Srinivasan}, booktitle={CRYPTO}, year={2016} }

The exact hardness of computing a Nash equilibrium is a fundamental open question in algorithmic game theory. This problem is complete for the complexity class PPAD. It is well known that problems in PPAD cannot be $$\mathrm {NP}$$ -complete unless $$\mathrm {NP}=\mathrm {coNP}$$ . Therefore, a natural direction is to reduce the hardness of PPAD to the hardness of problems used in cryptography.
Bitansky, Paneth, and Rosen [FOCS 2015] prove the hardness of PPAD assuming the existence of quasi…

## 73 Citations

Finding a Nash equilibrium is no easier than breaking Fiat-Shamir

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2019

It is shown that solving the END−OF−METERED−LINE problem is no easier than breaking the soundness of the Fiat-Shamir transformation when applied to the sumcheck protocol, and opens up the possibility of sampling moderately-sized games for which it is hard to find a Nash equilibrium.

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This work shows that hard-on-average TFNP problems can be based on the weak assumption that there exists a hard- on-average language in NP, in particular, this includes the assumption of the existence of one-way functions.

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We show that, relative to a random oracle, solving the End-of-Line problem (which is PPAD-complete) is no easier than computing the function f(N, x, T ) = x T mod N, where N is an n-bit RSA modulus,…

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The interplay between TFNP and OWFs is further explored and the first negative results are given, showing that there cannot exist constructions of average-case hard TFNP problem from OWFs with a certain type of simple black-box security reductions.

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A succinct non-interactive publicly-verifiable delegation scheme for any log-space uniform circuit under the sub-exponential Learning With Errors (LWE) assumption, and introduces a new cryptographic primitive: a lossy correlation-intractable hash function family.

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- Computer Science, MathematicsSODA
- 2016

The first hardness results for CLS are shown, showing instances for which any (computationally unbounded) randomized algorithm must perform exponentially many queries in order to find a local optimum and hardness for computationally bounded algorithms under cryptographic assumptions.

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The core observation behind the results is that the unique proofs property of incrementally-verifiable computations previously used to demonstrate hardness in PLS can be traded with a simple incremental completeness property.

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- Computer Science, Mathematics2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
- 2018

This work identifies the first PPP-complete problem without any circuit or Turing Machine given explicitly in the input, and thus answers a longstanding open question from [Papadimitriou1994].

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- 2017

This book provides strong evidence that even finding an approximate Nash equilibrium is intractable, and proves several intractability theorems for different settings (two-player games and many- player games) and models (computational complexity, query complexity, and communication complexity).

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