We revisit the problem of hardness amplification in $\NP$, as
recently studied by O'Donnell (STOC `02). We prove that if $\NP$
has a balanced function $f$ such that any circuit of size $s(n)$
fails to compute $f$ on a $1/\poly(n)$ fraction of inputs, then
$\NP$ has a function $f'$ such ... more >>>

Nisan (Combinatorica'92) constructed a pseudorandom generator for length $n$, width $n$ read-once branching programs (ROBPs) with error $\varepsilon$ and seed length $O(\log^2{n} + \log{n} \cdot \log(1/\varepsilon))$. A major goal in complexity theory is to reduce the seed length, hopefully, to the optimal $O(\log{n}+\log(1/\varepsilon))$, or to construct improved hitting sets, as ... more >>>

We present an explicit pseudorandom generator with seed length $\tilde{O}((\log n)^{w+1})$ for read-once, oblivious, width $w$ branching programs that can read their input bits in any order. This improves upon the work of Impaggliazzo, Meka and Zuckerman (FOCS'12) where they required seed length $n^{1/2+o(1)}$.

A central ingredient in our work ... more >>>

We construct pseudorandom generators of seed length $\tilde{O}(\log(n)\cdot \log(1/\epsilon))$ that $\epsilon$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom generators with seed length $\tilde{O}(\log(n) \cdot \mathrm{poly}(1/\epsilon))$. This is the first improvement for pseudorandom generators fooling width $3$ ROBPs since the work ... more >>>