All reports by Author Shuichi Hirahara:

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TR23-070
| 9th May 2023
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Shuichi Hirahara, Zhenjian Lu, Hanlin Ren#### Bounded Relativization

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TR23-037
| 28th March 2023
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Shuichi Hirahara#### Capturing One-Way Functions via NP-Hardness of Meta-Complexity

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TR23-035
| 22nd March 2023
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Shuichi Hirahara, Rahul Ilango, Zhenjian Lu, Mikito Nanashima, Igor Carboni Oliveira#### A Duality Between One-Way Functions and Average-Case Symmetry of Information

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TR23-026
| 15th March 2023
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Shuichi Hirahara, Nobutaka Shimizu#### Hardness Self-Amplification: Simplified, Optimized, and Unified

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TR22-164
| 20th November 2022
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Shuichi Hirahara, Mikito Nanashima#### Learning versus Pseudorandom Generators in Constant Parallel Time

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TR22-127
| 12th September 2022
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Eric Allender, Shuichi Hirahara, Harsha Tirumala#### Kolmogorov Complexity Characterizes Statistical Zero Knowledge

Revisions: 1

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TR22-119
| 24th August 2022
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Shuichi Hirahara#### NP-Hardness of Learning Programs and Partial MCSP

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TR22-108
| 18th July 2022
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Shuichi Hirahara, Nobutaka Shimizu#### Hardness Self-Amplification from Feasible Hard-Core Sets

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TR21-166
| 21st November 2021
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Lijie Chen, Shuichi Hirahara, Neekon Vafa#### Average-case Hardness of NP and PH from Worst-case Fine-grained Assumptions

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TR21-161
| 16th November 2021
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Shuichi Hirahara, Mikito Nanashima#### On Worst-Case Learning in Relativized Heuristica

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TR21-058
| 21st April 2021
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Shuichi Hirahara#### Average-Case Hardness of NP from Exponential Worst-Case Hardness Assumptions

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TR21-030
| 2nd March 2021
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Shuichi Hirahara, Rahul Ilango, Bruno Loff#### Hardness of Constant-round Communication Complexity

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TR21-010
| 11th February 2021
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Eric Allender, John Gouwar, Shuichi Hirahara, Caleb Robelle#### Cryptographic Hardness under Projections for Time-Bounded Kolmogorov Complexity

Revisions: 2

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TR20-143
| 16th September 2020
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Shuichi Hirahara#### Characterizing Average-Case Complexity of PH by Worst-Case Meta-Complexity

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TR20-103
| 9th July 2020
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Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, Yuichi Yoshida#### One-Tape Turing Machine and Branching Program Lower Bounds for MCSP

Revisions: 1

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TR20-050
| 18th April 2020
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Shuichi Hirahara#### Unexpected Hardness Results for Kolmogorov Complexity Under Uniform Reductions

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TR19-168
| 20th November 2019
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Igor Carboni Oliveira, Lijie Chen, Shuichi Hirahara, Ján Pich, Ninad Rajgopal, Rahul Santhanam#### Beyond Natural Proofs: Hardness Magnification and Locality

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TR19-025
| 28th February 2019
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Shuichi Hirahara, Osamu Watanabe#### On Nonadaptive Reductions to the Set of Random Strings and Its Dense Subsets

Revisions: 1

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TR18-138
| 10th August 2018
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Shuichi Hirahara#### Non-black-box Worst-case to Average-case Reductions within NP

Revisions: 1

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TR18-030
| 9th February 2018
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Shuichi Hirahara, Igor Carboni Oliveira, Rahul Santhanam#### NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits

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TR17-092
| 10th May 2017
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Shuichi Hirahara#### A Duality Between Depth-Three Formulas and Approximation by Depth-Two

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TR17-073
| 28th April 2017
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Eric Allender, Shuichi Hirahara#### New Insights on the (Non)-Hardness of Circuit Minimization and Related Problems

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TR15-198
| 30th November 2015
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Shuichi Hirahara, Osamu Watanabe#### Limits of Minimum Circuit Size Problem as Oracle

Revisions: 1

Shuichi Hirahara, Zhenjian Lu, Hanlin Ren

Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called *bounded relativization*. For a complexity class $C$, we say that a statement is *$C$-relativizing* if the statement holds relative ... more >>>

Shuichi Hirahara

A one-way function is a function that is easy to compute but hard to invert *on average*. We establish the first characterization of a one-way function by *worst-case* hardness assumptions, by introducing a natural meta-computational problem whose NP-hardness (and the worst-case hardness of NP) characterizes the existence of a one-way ... more >>>

Shuichi Hirahara, Rahul Ilango, Zhenjian Lu, Mikito Nanashima, Igor Carboni Oliveira

Symmetry of Information (SoI) is a fundamental property of Kolmogorov complexity that relates the complexity of a pair of strings and their conditional complexities. Understanding if this property holds in the time-bounded setting is a longstanding open problem. In the nineties, Longpré and Mocas (1993) and Longpré and Watanabe (1995) ... more >>>

Shuichi Hirahara, Nobutaka Shimizu

Strong (resp. weak) average-case hardness refers to the properties of a computational problem in which a large (resp. small) fraction of instances are hard to solve. We develop a general framework for proving hardness self-amplification, that is, the equivalence between strong and weak average-case hardness. Using this framework, we prove ... more >>>

Shuichi Hirahara, Mikito Nanashima

A polynomial-stretch pseudorandom generator (PPRG) in NC$^0$ (i.e., constant parallel time) is one of the most important cryptographic primitives, especially for constructing highly efficient cryptography and indistinguishability obfuscation. The celebrated work (Applebaum, Ishai, and Kushilevitz, SIAM Journal on Computing, 2006) on randomized encodings yields the characterization of sublinear-stretch pseudorandom generators ... more >>>

Eric Allender, Shuichi Hirahara, Harsha Tirumala

We show that a decidable promise problem has a non-interactive statistical zero-knowledge proof system if and only if it is randomly reducible to a promise problem for Kolmogorov-random strings, with a superlogarithmic additive approximation term. This extends recent work by Saks and Santhanam (CCC 2022). We build on this to ... more >>>

Shuichi Hirahara

A long-standing open question in computational learning theory is to prove NP-hardness of learning efficient programs, the setting of which is in between proper learning and improper learning. Ko (COLT'90, SICOMP'91) explicitly raised this open question and demonstrated its difficulty by proving that there exists no relativizing proof of NP-hardness ... more >>>

Shuichi Hirahara, Nobutaka Shimizu

We consider the question of hardness self-amplification: Given a Boolean function $f$ that is hard to compute on a $o(1)$-fraction of inputs drawn from some distribution, can we prove that $f$ is hard to compute on a $(\frac{1}{2} - o(1))$-fraction of inputs drawn from the same distribution? We prove hardness ... more >>>

Lijie Chen, Shuichi Hirahara, Neekon Vafa

What is a minimal worst-case complexity assumption that implies non-trivial average-case hardness of NP or PH? This question is well motivated by the theory of fine-grained average-case complexity and fine-grained cryptography. In this paper, we show that several standard worst-case complexity assumptions are sufficient to imply non-trivial average-case hardness ... more >>>

Shuichi Hirahara, Mikito Nanashima

A PAC learning model involves two worst-case requirements: a learner must learn all functions in a class on all example distributions. However, basing the hardness of learning on NP-hardness has remained a key challenge for decades. In fact, recent progress in computational complexity suggests the possibility that a weaker assumption ... more >>>

Shuichi Hirahara

A long-standing and central open question in the theory of average-case complexity is to base average-case hardness of NP on worst-case hardness of NP. A frontier question along this line is to prove that PH is hard on average if UP requires (sub-)exponential worst-case complexity. The difficulty of resolving this ... more >>>

Shuichi Hirahara, Rahul Ilango, Bruno Loff

How difficult is it to compute the communication complexity of a two-argument total Boolean function $f:[N]\times[N]\to\{0,1\}$, when it is given as an $N\times N$ binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard.

In this ... more >>>

Eric Allender, John Gouwar, Shuichi Hirahara, Caleb Robelle

A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT ... more >>>

Shuichi Hirahara

We exactly characterize the average-case complexity of the polynomial-time hierarchy (PH) by the worst-case (meta-)complexity of GapMINKT(PH), i.e., an approximation version of the problem of determining if a given string can be compressed to a short PH-oracle efficient program. Specifically, we establish the following equivalence:

DistPH is contained in ... more >>>

Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, Yuichi Yoshida

For a size parameter $s\colon\mathbb{N}\to\mathbb{N}$, the Minimum Circuit Size Problem (denoted by ${\rm MCSP}[s(n)]$) is the problem of deciding whether the minimum circuit size of a given function $f \colon \{0,1\}^n \to \{0,1\}$ (represented by a string of length $N := 2^n$) is at most a threshold $s(n)$. A ... more >>>

Shuichi Hirahara

Hardness of computing the Kolmogorov complexity of a given string is closely tied to a security proof of hitting set generators, and thus understanding hardness of Kolmogorov complexity is one of the central questions in complexity theory. In this paper, we develop new proof techniques for showing hardness of computing ... more >>>

Igor Carboni Oliveira, Lijie Chen, Shuichi Hirahara, Ján Pich, Ninad Rajgopal, Rahul Santhanam

Hardness magnification reduces major complexity separations (such as $EXP \not\subseteq NC^1$) to proving lower bounds for some natural problem $Q$ against weak circuit models. Several recent works [OS18, MMW19, CT19, OPS19, CMMW19, Oli19, CJW19a] have established results of this form. In the most intriguing cases, the required lower bound is ... more >>>

Shuichi Hirahara, Osamu Watanabe

We investigate the computational power of an arbitrary distinguisher for (not necessarily computable) hitting set generators as well as the set of Kolmogorov-random strings. This work contributes to (at least) two lines of research. One line of research is the study of the limits of black-box reductions to some distributional ... more >>>

Shuichi Hirahara

There are significant obstacles to establishing an equivalence between the worst-case and average-case hardness of NP: Several results suggest that black-box worst-case to average-case reductions are not likely to be used for reducing any worst-case problem outside coNP to a distributional NP problem.

This paper overcomes the barrier. We ... more >>>

Shuichi Hirahara, Igor Carboni Oliveira, Rahul Santhanam

The Minimum Circuit Size Problem (MCSP) asks for the size of the smallest boolean circuit that computes a given truth table. It is a prominent problem in NP that is believed to be hard, but for which no proof of NP-hardness has been found. A significant number of works have ... more >>>

Shuichi Hirahara

We establish an explicit link between depth-3 formulas and one-sided approximation by depth-2 formulas, which were previously studied independently. Specifically, we show that the minimum size of depth-3 formulas is (up to a factor of n) equal to the inverse of the maximum, over all depth-2 formulas, of one-sided-error correlation ... more >>>

Eric Allender, Shuichi Hirahara

The Minimum Circuit Size Problem (MCSP) and a related problem (MKTP) that deals with time-bounded Kolmogorov complexity are prominent candidates for NP-intermediate status. We show that, under very modest cryptographic assumptions (such as the existence of one-way functions), the problem of approximating the minimum circuit size (or time-bounded Kolmogorov complexity) ... more >>>

Shuichi Hirahara, Osamu Watanabe

The Minimum Circuit Size Problem (MCSP) is known to be hard for statistical zero knowledge via a BPP-reduction (Allender and Das, 2014), whereas establishing NP-hardness of MCSP via a polynomial-time many-one reduction is difficult (Murray and Williams, 2015) in the sense that it implies ZPP $\neq$ EXP, which is a ... more >>>