Polynomial-Time Pseudodeterministic Construction of Primes

Chen, Lijie; Lu, Zhenjian; Oliveira, Igor C.; Ren, Hanlin; Santhanam, Rahul

doi:10.1109/focs57990.2023.00074

2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) 2023

DOI: 10.1109/focs57990.2023.00074

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Polynomial-Time Pseudodeterministic Construction of Primes

Lijie Chen,

Zhenjian Lu,

Igor C. Oliveira

et al.

Abstract: How to cite:Please refer to published version for the most recent bibliographic citation information. If a published version is known of, the repository item page linked to above, will contain details on accessing it.

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Cited by 4 publications

References 41 publications

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Distinguishing, Predicting, and Certifying: On the Long Reach of Partial Notions of Pseudorandomness

Li,

Pyne,

Tell

2024

2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)

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Distinguishing, Predicting, and Certifying: On the Long Reach of Partial Notions of Pseudorandomness

Li,

Pyne,

Tell

2024

2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)

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On the Complexity of Avoiding Heavy Elements

Lu,

Oliveira,

Ren

et al. 2024

2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)

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We introduce and study the following natural total search problem, which we call the heavy element avoidance (Heavy Avoid) problem: for a distribution on N bits specified by a Boolean circuit sampling it, and for some parameter δ(N ) ≥ 1/poly(N ) fixed in advance, output an N -bit string that has probability less than δ(N ). We show that the complexity of Heavy Avoid is closely tied to frontier open questions in complexity theory about uniform randomized lower bounds and derandomization. Among other results, we show:1. For a wide range of circuit classes C, including ACC 0 , TC 0 , NC 1 and general Boolean circuits, EXP does not have uniform randomized C-circuits if and only if Heavy Avoid for uniform implicit C-samplers has efficient deterministic algorithms infinitely often. This gives the first algorithmic characterization of lower bounds for EXP against uniform randomized low-depth circuits. We show similar algorithmic characterizations for lower bounds in PSPACE, NP and EXP NP .2. Unconditionally, there are polynomial-time pseudodeterministic algorithms that work infinitely often for several variants of Heavy Avoid, such as for uniform samplers of small randomness complexity. In contrast, the existence of a similar algorithm that solves Heavy Avoid for arbitrary polynomial-time samplers would solve a long-standing problem about hierarchies for probabilistic time.3. If there is a time and depth efficient deterministic algorithm for Heavy Avoid, then BPP = P. Without the depth-efficiency requirement in the assumption, we still obtain a non-trivial form of infinitely-often deterministic simulation of randomized algorithms. These results are shown using non-black-box reductions, and we argue that the use of non-black-box reductions is essential here.

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Hardness of Range Avoidance and Remote Point for Restricted Circuits via Cryptography

Chen,

2024

Proceedings of the 56th Annual ACM Symposium on Theory of Computing

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Polynomial-Time Pseudodeterministic Construction of Primes

Abstract: How to cite:Please refer to published version for the most recent bibliographic citation information. If a published version is known of, the repository item page linked to above, will contain details on accessing it.

Cited by 4 publications

References 41 publications

Distinguishing, Predicting, and Certifying: On the Long Reach of Partial Notions of Pseudorandomness

Distinguishing, Predicting, and Certifying: On the Long Reach of Partial Notions of Pseudorandomness

On the Complexity of Avoiding Heavy Elements

Hardness of Range Avoidance and Remote Point for Restricted Circuits via Cryptography

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