## Small-Bias Probability Spaces: Efficient Constructions and Applications (1993)

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Venue: | SIAM J. Comput |

Citations: | 253 - 14 self |

### BibTeX

@ARTICLE{Naor93small-biasprobability,

author = {Joseph Naor and Moni Naor},

title = {Small-Bias Probability Spaces: Efficient Constructions and Applications},

journal = {SIAM J. Comput},

year = {1993},

volume = {22},

pages = {838--856}

}

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### Abstract

We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called ffl-biased random variables. The number of random bits needed to generate the random variables is O(log n + log 1 ffl ). Thus, if ffl is polynomially small, then the size of the sample space is also polynomial. Random variables that are ffl-biased can be used to construct "almost" k-wise independent random variables where ffl is a function of k. These probability spaces have various applications: 1. Derandomization of algorithms: many randomized algorithms that require only k- wise independence of their random bits (where k is bounded by O(log n)), can be derandomized by using ffl-biased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive testing of combinatorial circui...