Abstract:
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 circuits. We prove a \emph{fully uniform} version of the Direct-Product Theorem with information-theoretically \emph{optimal} parameters, up to constant factors. Namely, we show that for given $k$ and $\epsilon$, there is an efficient randomized algorithm $A$ with the following property. Given a circuit $C$ that computes $f^k$ on at least $\epsilon$ fraction of inputs, the algorithm $A$ outputs with probability at least $3/4$ a list of $O(1/\epsilon)$ circuits such that at least one of the circuits on the list computes $f$ on more than $1-\delta$ fraction of inputs, for $\delta = O((\log 1/\epsilon)/k)$; moreover, each output circuit is an $\AC^0$ circuit (of size $\poly(n,k,\log 1/\delta,1/\epsilon)$), with oracle access to the circuit $C$. Using the Goldreich-Levin decoding algorithm~\cite{GL89}, we also get a \emph{fully uniform} version of Yao's XOR Lemma~\cite{Yao} with \emph{optimal} parameters, up to constant factors. Our results simplify and improve those in~\cite{IJK06}. Our main result may be viewed as an efficient approximate, local, list-decoding algorithm for direct-product codes (encoding a function by its values on all $k$-tuples) with optimal parameters. We generalize it to a family of ``derandomized" direct-product codes, which we call {\em intersection codes}, where the encoding provides values of the function only on a subfamily of $k$-tuples. The quality of the decoding algorithm is then determined by sampling properties of the sets in this family and their intersections. As a direct consequence of this generalization we obtain the first derandomized direct product result in the uniform setting, allowing hardness amplification with only constant (as opposed to a factor of $k$) increase in the input length. Finally, this general setting naturally allows the decoding of concatenated codes, which further yields nearly optimal derandomized amplification.