Separating the polynomial-time hierarchy by oracles
Proc. 26th annual symposium on Foundations of computer science
Computational limitations of small-depth circuits
Computational limitations of small-depth circuits
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
A hard-core predicate for all one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
n&OHgr;(logn) lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom
Information Processing Letters
Threshold circuits of bounded depth
Journal of Computer and System Sciences
Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs
Journal of Computer and System Sciences
Journal of Computer and System Sciences
Lower bounds for sampling algorithms for estimating the average
Information Processing Letters
BPP has subexponential time simulations unless EXPTIME has publishable proofs
Computational Complexity
When do extra majority gates help?: polylog(N) majority gates are equivalent to one
Computational Complexity - Special issue on circuit complexity
On the hardness of computing the permanent of random matrices
Computational Complexity
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
SIAM Journal on Computing
Pseudorandom generators without the XOR lemma
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Extractors and pseudorandom generators
Journal of the ACM (JACM)
Randomness vs time: derandomization under a uniform assumption
Journal of Computer and System Sciences
Information Theory: Coding Theorems for Discrete Memoryless Systems
Information Theory: Coding Theorems for Discrete Memoryless Systems
Boosting and Hard-Core Set Construction
Machine Learning
Hiding Instances in Multioracle Queries
STACS '90 Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science
Hard Sets and Pseudo-random Generators for Constant Depth Circuits
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
On the Derandomization of Constant Depth Circuits
APPROX '01/RANDOM '01 Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 5th International Workshop on Randomization and Approximation Techniques in Computer Science: Approximation, Randomization and Combinatorial Optimization
Hard-core distributions for somewhat hard problems
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
A theorem on probabilistic constant depth Computations
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Lower Bounds for Approximations by Low Degree Polynomials Over Z_m
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
List-Decoding Using The XOR Lemma
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Number-theoretic constructions of efficient pseudo-random functions
Journal of the ACM (JACM)
Hardness amplification within NP
Journal of Computer and System Sciences - Special issue on computational complexity 2002
Simple extractors for all min-entropies and a new pseudorandom generator
Journal of the ACM (JACM)
On uniform amplification of hardness in NP
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
The complexity of constructing pseudorandom generators from hard functions
Computational Complexity
On Constructing Parallel Pseudorandom Generators from One-Way Functions
CCC '05 Proceedings of the 20th Annual IEEE Conference on Computational Complexity
On the Complexity of Hardness Amplification
CCC '05 Proceedings of the 20th Annual IEEE Conference on Computational Complexity
Using Nondeterminism to Amplify Hardness
SIAM Journal on Computing
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Approximately List-Decoding Direct Product Codes and Uniform Hardness Amplification
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The complexity of hardness amplification and derandomization
The complexity of hardness amplification and derandomization
Note: Improved hardness amplification in NP
Theoretical Computer Science
Verifying and decoding in constant depth
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Pseudorandomness for Approximate Counting and Sampling
Computational Complexity
Pseudorandom Bits for Constant-Depth Circuits with Few Arbitrary Symmetric Gates
SIAM Journal on Computing
Foundations and Trends® in Theoretical Computer Science
On Worst-Case to Average-Case Reductions for NP Problems
SIAM Journal on Computing
Pseudorandomness and Average-Case Complexity Via Uniform Reductions
Computational Complexity
Hardness amplification proofs require majority
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Hardness Amplification within NP against Deterministic Algorithms
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Theory and application of trapdoor functions
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
The Complexity of Local List Decoding
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Bit-probe lower bounds for succinct data structures
Proceedings of the forty-first annual ACM symposium on Theory of computing
Concentration of Measure for the Analysis of Randomized Algorithms
Concentration of Measure for the Analysis of Randomized Algorithms
On Approximate Majority and Probabilistic Time
Computational Complexity
On the Power of Small-Depth Computation
Foundations and Trends® in Theoretical Computer Science
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
Cell-probe lower bounds for succinct partial sums
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized
SIAM Journal on Computing
On hardness amplification of one-way functions
TCC'05 Proceedings of the Second international conference on Theory of Cryptography
Hardness amplification via space-efficient direct products
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
Impossibility results on weakly black-box hardness amplification
FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
On the complexity of hard-core set constructions
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Query complexity in errorless hardness amplification
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
On beating the hybrid argument
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
On the power of nonuniformity in proofs of security
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Natural proofs versus derandomization
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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Hardness amplification is the fundamental task of converting a $\delta$-hard function $f:\{0,1\}^n\to\{0,1\}$ into a $(1/2-\epsilon)$-hard function $\mathit{Amp}(f)$, where $f$ is $\gamma$-hard if small circuits fail to compute $f$ on at least a $\gamma$ fraction of the inputs. In this paper we study the complexity of black-box proofs of hardness amplification. A class of circuits $\mathcal{D}$ proves a hardness amplification result if for any function $h$ that agrees with $\mathit{Amp}(f)$ on a $1/2+\epsilon$ fraction of the inputs there exists an oracle circuit $D\in\mathcal{D}$ such that $D^h$ agrees with $f$ on a $1-\delta$ fraction of the inputs. We focus on the case where every $D\in\mathcal{D}$ makes nonadaptive queries to $h$. This setting captures most hardness amplification techniques. We prove two main results: (1) The circuits in $\mathcal{D}$ “can be used” to compute the majority function on $1/\epsilon$ bits. In particular, when $\epsilon\leq1/\log^{\omega(1)}n$, $\mathcal{D}$ cannot consist of oracle circuits that have unbounded fan-in, size $\mathrm{poly}(n)$, and depth $O(1)$. (2) The circuits in $\mathcal{D}$ must make $\Omega\left(\log(1/\delta)/\epsilon^2\right)$ oracle queries. Both our bounds on the depth and on the number of queries are tight up to constant factors. Our results explain why hardness amplification techniques have failed to transform known lower bounds against constant-depth circuit classes into strong average-case lower bounds. Our results reveal a contrast between Yao's XOR lemma ($\mathit{Amp}(f):=f(x_1)\oplus\cdots\oplus f(x_t)\in\{0,1\}$) and the direct-product lemma ($\mathit{Amp}(f):=f(x_1)\circ\cdots\circ f(x_t)\in\{0,1\}^t$; here $\mathit{Amp}(f)$ is non-Boolean). Our results (1) and (2) apply to Yao's XOR lemma, whereas known proofs of the direct-product lemma violate both (1) and (2). One of our contributions is a new technique for handling “nonuniform” reductions, i.e., the case when $\mathcal{D}$ contains many circuits.