Explicit resilient functions matching AjtaiLinial
Abstract
A Boolean function on n variables is qresilient if for any subset of at most q variables, the function is very likely to be determined by a uniformly random assignment to the remaining nq variables; in other words, no coalition of at most q variables has significant influence on the function. Resilient functions have been extensively studied with a variety of applications in cryptography, distributed computing, and pseudorandomness. The best known balanced resilient function on n variables due to Ajtai and Linial ([AL93]) is Omega(n/(log^2 n))resilient. However, the construction of Ajtai and Linial is by the probabilistic method and does not give an efficiently computable function. In this work we give an explicit monotone depth three almostbalanced Boolean function on n bits that is Omega(n/(log^2 n))resilient matching the work of Ajtai and Linial. The best previous explicit construction due to Meka [Meka09] (which only gives a logarithmic depth function) and Chattopadhyay and Zuckermman [CZ15] were only n^{1c}resilient for any constant c < 1. Our construction and analysis are motivated by (and simplifies parts of) the recent breakthrough of [CZ15] giving explicit twosources extractors for polylogarithmic minentropy; a key ingredient in their result was the construction of explicit constantdepth resilient functions. An important ingredient in our construction is a new randomness optimal oblivious sampler which preserves moment generating functions of sums of variables and could be useful elsewhere.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 arXiv:
 arXiv:1509.00092
 Bibcode:
 2015arXiv150900092M
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms