Elements of information theory
Elements of information theory
Journal of Computer and System Sciences
Learning Polynomials with Queries: The Highly Noisy Case
SIAM Journal on Discrete Mathematics
Pseudorandom generators without the XOR lemma
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Hardness amplification within NP
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
The Nonstochastic Multiarmed Bandit Problem
SIAM Journal on Computing
Boosting and Hard-Core Set Construction
Machine Learning
Hard-core distributions for somewhat hard problems
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
List-Decoding Using The XOR Lemma
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
On uniform amplification of hardness in NP
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Key agreement from weak bit agreement
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Using Nondeterminism to Amplify Hardness
SIAM Journal on Computing
Theory and application of trapdoor functions
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Conditional Computational Entropy, or Toward Separating Pseudoentropy from Compressibility
EUROCRYPT '07 Proceedings of the 26th annual international conference on Advances in Cryptology
The uniform hardcore lemma via approximate Bregman projections
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Extracting Computational Entropy and Learning Noisy Linear Functions
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Deterministic extractors for independent-symbol sources
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
On the complexity of hard-core set constructions
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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The seminal hardcore lemma of Impagliazzo states that for any mildly-hard Boolean function f, there is a subset of input, called the hardcore set, on which the function is extremely hard, almost as hard as a random Boolean function. This implies that the output distribution of f given a random input looks like a distribution with some statistical randomness. Can we have something similar for hard functions with several output bits? Can we say that the output distribution of such a general function given a random input looks like a distribution containing several bits of randomness? If so, one can simply apply any statistical extractor to extract computational randomness from the output of f. However, the conventional wisdom tells us to apply extractors with some additional reconstruction property, instead of just any extractor. Does this mean that there is no analogous hardcore lemma for general functions? We show that a general hard function does indeed have some kind of hardcore set, but it comes with the price of a security loss which is proportional to the number of output values. More precisely, consider a hard function f : {0, 1}n → [V ] = {1, . . ., V } such that any circuit of size s can only compute f correctly on at most 1/L(1 - γ) fraction of inputs, for some L ∈ [1, V - 1] and γ ∈ (0, 1). Then we show that for some I ⊆ [V] with |I| = L + 1, there exists a hardcore set HI ⊆ f-1(I) with density γ(V L+1) such that any circuit of some size s′ can only compute f correctly on at most 1+ε/L+1 fraction of inputs in HI. Here, s′ is smaller than s by some poly(V, 1/ε, log(1/γ)) factor, which results in a security loss of such a factor. We show that it is basically impossible to guarantee a much larger hardcore set or a much smaller security loss. Finally, we show how our hardcore lemma can be used for extracting computational randomness from general hard functions.