The classical Direct-Product Theorem for circuits says
that if a Boolean function $f:\{0,1\}^n\to\{0,1\}$ is somewhat hard
to compute on average by small circuits, then the corresponding
$k$-wise direct product function
$f^k(x_1,\dots,x_k)=(f(x_1),\dots,f(x_k))$ (where each
$x_i\in\{0,1\}^n$) is significantly harder to compute on average by
slightly smaller ... more >>>