## Better pseudorandom generators from milder pseudorandom restrictions

by Parikshit Gopalan, Raghu Meka, Omer Reingold, Luca Trevisan, Salil Vadhan

#### Oded's comments

This is another recent work (cf. Choise Nr 103) that uses pseudorandom restrictions to obtain improved pseudorandomn generators; this time for combinatorial rectangles and read-once CNFs.

#### The original abstract

We present an iterative approach to constructing pseudorandom generators, based on the repeated application of mild pseudorandom restrictions. We use this template to construct pseudorandom generators for combinatorial rectangles and read-once CNFs and a hitting set generator for width-3 branching programs, all of which achieve near optimal seed-length even in the low-error regime: We get seed-length $\tilde{O}(\log (n/\epsilon))$ for error $\epsilon$. Previously, only constructions with seed-length $O(\log^{3/2} n)$ or $O(\log^2 n)$ were known for these classes with polynomially small error.

The (pseudo)random restrictions we use are milder than those typically used for proving circuit lower bounds in that we only set a constant fraction of the bits at a time. While such restrictions do not simplify the functions drastically, we show that they can be derandomized using small-bias spaces.

See ECCC TR12-123.

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