An improved derandomization of the switching lemma

We prove a new derandomization of Håstad’s switching lemma, showing how to efficiently generate restrictions satisfying the switching lemma for DNF or CNF formulas of size m using only O (log m ) random bits. Derandomizations of the switching lemma have been useful in many works as a key building-block for constructing objects which are in some way provably-pseudorandom with respect to AC 0 -circuits. Here, we use our new derandomization to give an improved analysis of the pseudorandom generator of Trevisan and Xue for AC 0 -circuits (CCC’13): we show that the generator ε-fools size- m , depth- D circuits with n -bit inputs using only O (log( m /ε) D · log n ) random bits. In particular, we obtain (modulo the loglog-factors hidden in the O -notation) a dependence on m /ε which is best-possible with respect to currently-known AC 0 -circuit lower bounds.

Authors

• Zander Kelley

Keywords

• pseudorandomness
• switching lemma