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 NP-complete unless NP = 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-polynomially hard indistinguishability obfuscation and sub-exponentially hard one-way functions. This leaves open the possibility of basing PPAD hardness on simpler, polynomially hard, computational assumptions. We make further progress in this direction and reduce PPAD hardness directly to polynomially hard assumptions. Our first result proves hardness of PPAD assuming the existence of polynomially hard indistinguishability obfuscation (iO) and one-way permutations. While this improves upon Bitansky et al.'s work, it does not give us a reduction to simpler, polynomially hard computational assumption because constructions of iO inherently seems to require assumptions with sub-exponential hardness. In contrast, public key functional encryption is a much simpler primitive and does not suffer from this drawback. Our second result shows that PPAD hardness can be based on polynomially hard compact public key functional encryption and one-way permutations. Our results further demonstrate the power of polynomially hard compact public key functional encryption which is believed to be weaker than indistinguishability obfuscation. Our techniques are general and we expect them to have various applications.
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