Hardness of Continuous Local Search: Query Complexity and Cryptographic Lower Bounds

Hubáček, Pavel; Yogev, Eylon

doi:10.1137/17m1118014

Cited by 22 publications

(34 citation statements)

References 29 publications

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“…We establish the containment in CLS through a reduction to E -O -M -L (EOML), a total search problem that was recently introduced by Hubáček and Yogev [8] who also showed that it is in CLS. In EOML we are given a source in a directed graph with vertices of in-degree and out-degree at most one, and the task is to find a sink or a source different from the given trivial source.…”

Section: Our Resultsmentioning

confidence: 99%

ARRIVAL: Next Stop in CLS

Gärtner¹,

Hansen²,

Hubáček³

et al. 2018

Preprint

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We study the computational complexity of A , a zero-player game on n-vertex switch graphs introduced by Dohrau, Gärtner, Kohler, Matoušek, and Welzl. They showed that the problem of deciding termination of this game is contained in NP ∩ coNP. Karthik C. S. recently introduced a search variant of A and showed that it is in the complexity class PLS. In this work, we significantly improve the known upper bounds for both the decision and the search variants of A . First, we resolve a question suggested by Dohrau et al. and show that the decision variant of A is in UP ∩ coUP. Second, we prove that the search variant of A is contained in CLS. Third, we give a randomized O(1.4143 n )-time algorithm to solve both variants.Our main technical contributions are (a) an efficiently verifiable characterization of the unique witness for termination of the A game, and (b) an efficient way of sampling from the state space of the game. We show that the problem of finding the unique witness is contained in CLS, whereas it was previously conjectured to be FPSPACE-complete. The efficient sampling procedure yields the first algorithm for the problem that has expected runtime O(c n ) with c < 2.

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Section: Our Resultsmentioning

confidence: 99%

ARRIVAL: Next Stop in CLS

Gärtner¹,

Hansen²,

Hubáček³

et al. 2018

Preprint

Self Cite

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show abstract

“…Building on [AKV04], Bitanski, Paneth and Rosen [BPR15] show that PPAD is hard on average assuming sub-exponentially secure indistinguishability obfuscation. Hubá ek and Yogev [HY17] extended this result to CLS, a subclass of PPAD.…”

Section: Introductionmentioning

confidence: 81%

Delegation with Updatable Unambiguous Proofs and PPAD-Hardness

Kalai

Paneth

Yang

2020

Advances in Cryptology – CRYPTO 2020

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In this work, we construct an updatable and unambiguous delegation scheme based on the decisional assumption on bilinear groups introduced by Kalai, Paneth and Yang [STOC 2019]. Using this delegation scheme, we show PPAD-hardness (and hence the hardness of computing Nash equilibria) based on the quasi-polynomial hardness of this bilinear group assumption and any hard language that is decidable in quasi-polynomial time and polynomial space.The delegation scheme is for super-polynomial time deterministic computations and is publicly verifiable and non-interactive in the common reference string (CRS) model. It is updatable meaning that given a proof for the statement that a Turing machine reaches some configuration C in T steps, it is efficient to update it into a proof for the statement that the machine reaches the next configuration C' in T+1 steps. It is unambiguous meaning that it is hard to produce two different proofs for the same statement.

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“…• Fully homomorphic encryption scheme for unbounded-depth polynomial size circuits (without relying on circular security), assuming slighly superpolynomial hardness of the four assumptions [50]. • PPAD hardness [1,26,67,86,98].…”

Section: Corollary 13 (Informal)mentioning

confidence: 99%

Indistinguishability obfuscation from well-founded assumptions

Jain

Lin

Sahai

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

155

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Indistinguishability obfuscation, introduced by [Barak et. al. Crypto 2001], aims to compile programs into unintelligible ones while preserving functionality. It is a fascinating and powerful object that has been shown to enable a host of new cryptographic goals and beyond. However, constructions of indistinguishability obfuscation have remained elusive, with all other proposals relying on heuristics or newly conjectured hardness assumptions.In this work, we show how to construct indistinguishability obfuscation from subexponential hardness of four well-founded assumptions. We prove: Informal Theorem: Let 𝜏 ∈ (0, ∞), 𝛿 ∈ (0, 1), 𝜖 ∈ (0, 1) be arbitrary constants. Assume sub-exponential security of the following assumptions:-the Learning With Errors (LWE) assumption with subexponential modulus-to-noise ratio 2 𝑘 𝜖 and noises of magnitude polynomial in 𝑘, where 𝑘 is the dimension of the LWE secret, -the Learning Parity with Noise (LPN) assumption over general prime fields Z 𝑝 with polynomially many LPN samples and error rate 1/ℓ 𝛿 , where ℓ is the dimension of the LPN secret, -the existence of a Boolean Pseudo-Random Generator (PRG) in NC 0 with stretch 𝑛 1+𝜏 , where 𝑛 is the length of the PRG seed, -the Decision Linear (DLIN) assumption on symmetric bilinear groups of prime order. Then, (subexponentially secure) indistinguishability obfuscation for all polynomial-size circuits exists. Further, assuming only polynomial security of the aforementioned assumptions, there exists collusion resistant public-key functional encryption for all polynomial-size circuits. CCS CONCEPTS• Theory of computation → Cryptographic primitives.

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Hardness of Continuous Local Search: Query Complexity and Cryptographic Lower Bounds

Cited by 22 publications

References 29 publications

ARRIVAL: Next Stop in CLS

ARRIVAL: Next Stop in CLS

Delegation with Updatable Unambiguous Proofs and PPAD-Hardness

Indistinguishability obfuscation from well-founded assumptions

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