Ragesh Jaiswal

Approximately List-Decoding Direct Product Codes and Uniform Hardness Amplification

We consider the problem of approximately locally listdecoding direct product codes. For a parameter k, the kwise direct product encoding of an N-bit message msg is an N^{k}-length string over the alphabet {0, 1}^k indexed by ktuples (i_1, . . . , i_k) in {1, . . . ,N}^k so that the symbol at position (i_1, . . . , i_k) of the codeword is msg(i_1) . . . msg(i_k). Such codes arise naturally in the context of hardness amplification of Boolean functions via the Direct Product Lemma (and the closely related Yao�s XOR Lemma), where typically k ll N (e.g., k = poly logN). We describe an efficient randomized algorithm for approximate local list-decoding of direct product codes. Given access to a word which agrees with the k-wise direct product encoding of some message msg in at least an in fraction of positions, our algorithm outputs a list of poly(1/in) Boolean circuits computing N-bit strings (viewed as truth tables of logN-variable Boolean functions) such that at least one of them agrees with msg in at least 1 - delta fraction of positions, for delta = O(k^{-0.1}), provided that in =Omega(poly(1/k)); the running time of the algorithm is polynomial in logN and 1/in. When in ge e-^{k^{alpha} } for a certain constant alpha > 0, we get a randomized approximate listdecoding algorithm that runs in time quasi-polynomial in 1/in (i.e., (1/in)^{poly log 1/in}). By concatenating the k-wise direct product codes with Hadamard codes, we obtain locally list-decodable codes over the binary alphabet, which can be efficiently approximately list-decoded from fewer than 1/2 - in fraction of corruptions as long as in = Omega(poly(1/k)). As an immediate application, we get uniform hardness amplification for P^{NP_parallel} , the class of languages reducible to NP through one round of parallel oracle queries: If there is a language in P^{NP_parallel} that cannot be decided by any BPP algorithm on more that 1 - 1/n^{Omega(1)} fraction of inputs, then there is another language in P^{NP_parallel} that cannot be decided by any BPP algorithm on more than 1/2 + 1/n^{omega(1)} fraction of inputs.

Uniform direct product theorems: simplified, optimized, and derandomized

The classical Direct-Product Theorem for circuits says that if a Boolean function f: {0,1}n -> {0,1} is somewhat hard to compute on average by small circuits, then the corresponding k-wise direct product function fk(x1,...,xk)=(f(x1),...,f(xk)) (where each xi -> {0,1}n) is significantly harder to compute on average by slightly smaller circuits. We prove a fully uniform version of the Direct-Product Theorem with information-theoretically optimal parameters, up to constant factors. Namely, we show that for given k and ε, there is an efficient randomized algorithm A with the following property. Given a circuit C that computes fk on at least ε fraction of inputs, the algorithm A outputs with probability at least 3/4 a list of O(1/ε) circuits such that at least one of the circuits on the list computes f on more than 1-δ fraction of inputs, for δ = O((log 1/ε)/k). Moreover, each output circuit is an AC0 circuit (of size poly(n,k,log 1/δ,1/ε)), with oracle access to the circuit C. Using the Goldreich-Levin decoding algorithm [5], we also get a fully uniform version of Yaos XOR Lemma [18] with optimal parameters, up to constant factors. Our results simplify and improve those in [10]. Our main result may be viewed as an efficient approximate, local, list-decoding algorithm for direct-product codes (encoding a function by its values on all k-tuples) with optimal parameters. We generalize it to a family of derandomized direct-product codes, which we call intersection codes, where the encoding provides values of the function only on a subfamily of k-tuples. The quality of the decoding algorithm is then determined by sampling properties of the sets in this family and their intersections. As a direct consequence of this generalization we obtain the first derandomized direct product result in the uniform setting, allowing hardness amplification with only constant (as opposed to a factor of k) increase in the input length. Finally, this general setting naturally allows the decoding of concatenated codes, which further yields nearly optimal derandomized amplification.

Bounded Independence Fools Halfspaces

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A Simple D2-Sampling Based PTAS for k-Means and Other Clustering Problems

pmo^n

Chernoff-type direct product theorems

that is

Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized

k

Security Amplification for Interactive Cryptographic Primitives

-wise independent fools any halfspace (a.k.a.~threshold)

Chernoff-Type Direct Product Theorems

h : pmo^n to pmo

Improved analysis of D 2 -sampling based PTAS for k-means and other clustering problems

, i.e., any function of the form

Analysis of k-Means++ for Separable Data

h(x) = sign(sum_{i = 1}^n w_i x_i -theta)