## Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized

by Russell Impagliazzo, Ragesh Jaiswal, Valentine Kabanets, and Avi Wigderson

What is striking in this version of the Direct Product Theorem is that it does not require samples of (input,value) pairs for the basic function (denoted $f$). Instead, the reduction (of strongly approximating $f$ to weakly approximating $f^k$) uses the oracle to generate such samples. Actually, the reduction makes a single query and based on it produces, with small probability, an oracle machine that strongly approximates $f$ (when given oracle access to a weak approximator of $f^k$).
Specifically, for $k'=k/2$, the reduction selects uniformly a $k'$-subset $A$ of the domain, and generates a machine $M_{A,V}$, where $V$ is the value obtained from the oracle when querying it on a random $k$-set that contains $A$. On input $x$ and oracle access to $F$ (which weakly approximates $f^k$), the machine $C_{A,V}$ samples random $k$-sets $B$ that contain $A$ and $x$, and returns $F(B)_x$ for the first $V$ that satisfies $F(B)_A=V$. There is also a derandomized version, which I did not study yet (shame on me!).
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 $C$.
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.