## Uniform direct product theorems: Simplified, optimized, and derandomized

### Abstract

The classical Direct-Product Theorem for circuits says that if a Boolean function f:{0,1}n→{0,1} is somewhat hard to compute on average by small circuits, then the corresponding k-wise direct product function fk(x1, ..., xk)=(f(x1), ...,f(xk)) (where each xi∈{0,1}n) is significantly harder to compute on average by slightly smaller circuits. We prove a fully uniform version of the Direct-Product Theorem with information-theoretically optimal parameters, up to constant factors. Namely, we show that for given k and ε, there is an efficient randomized algorithm A with the following property. Given a circuit C that computes fk on at least ε fraction of inputs, the algorithm A outputs with probability at least 3/4 a list of O(1/ε) circuits such that at least one of the circuits on the list computes f on more than 1-δ fraction of inputs, for δ= O((log 1/ε)/k); moreover, each output circuit is an AC0 circuit (of size poly(n,k,log 1/δ,1/ε)), with oracle access to the circuit C.

Using the Goldreich-Levin decoding algorithm, we also get a fully uniform version of Yao's XOR Lemma with optimal parameters, up to constant factors. Our results simplify and improve those in Impagliazzo et al.

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 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.

### Versions

• journal version SIAM Journal on Computing, 39(4):1637-1665, 2010.
• extended abstract in Proceedings of the Fortieth ACM Symposium on Theory of Computing (STOC'08), pages 579-588, 2008.