For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$.

When $p \equiv 3$ mod $4$, it is well known that $f(X) = X^{(p+1)/4}$ computes square ... more >>>

We study hardness amplification in the context of two well-known "moderate" average-case hardness results for $\mathrm{AC}^0$ circuits. First, we investigate the extent to which $\mathrm{AC}^0$ circuits of depth $d$ can approximate $\mathrm{AC}^0$ circuits of some larger depth $d + k$. The case $k = 1$ is resolved by Håstad, Rossman, ... more >>>

The Perebor (Russian for “brute-force search”) conjectures, which date back to the 1950s and 1960s are some of the oldest conjectures in complexity theory. The conjectures are a stronger form of the NP ? = P conjecture (which they predate) and state that for “meta-complexity” problems, such as the Time-bounded ... more >>>