We show that derandomizing Polynomial Identity Testing is essentially equivalent to proving arithmetic circuit lower bounds for NEXP. More precisely, we prove that if one can test in polynomial time (or even nondeterministic subexponential time, infinitely often) whether a given arithmetic circuit over integers computes an identically zero polynomial, then either (i) NEXP ⊂ P/poly or (ii) Permanent is not computable by polynomial-size arithmetic circuits. We also prove a (partial) converse: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic circuit of polynomially bounded degree computes an identically zero polynomial. Since Polynomial Identity Testing is a coRP problem, we obtain the following corollary: If RP = P (or even coRP ⊆ >0 NTIME(2 n ), infinitely often), then NEXP is not computable by polynomial-size arithmetic circuits. Thus establishing that RP = coRP or BPP = P would require proving superpolynomial lower bounds for Boolean or arithmetic circuits. We also show that any derandomization of RNC would yield new circuit lower bounds for a language in NEXP. We also prove unconditionally that NEXP RP does not have polynomialsize Boolean or arithmetic circuits. Finally, we show that NEXP ⊂ P/poly if both BPP = P and low-degree testing is in P; here low-degree testing is the problem of checking whether a given Boolean circuit computes a function that is close to some low-degree polynomial over a finite field.
We show that derandomizing Polynomial Identity Testing is, essentially, equivalent to proving circuit lower bounds for NEXP. More precisely, we prove that if one can test in polynomial time (or, even, nondeterministic subexponential time, infinitely often) whether a given arithmetic circuit over integers computes an identically zero polynomial, then either (i) NEXP ⊂ P/poly or (ii) Permanent is not computable by polynomial-size arithmetic circuits. We also prove a (partial) converse: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic formula computes an identically zero polynomial.Since Polynomial Identity Testing is a coRP problem, we obtain the following corollary: If RP = P (or, even, coRP ⊆ ∩ >0NTIME(2 n ), infinitely often), then NEXP is not computable by polynomial-size arithmetic circuits. Thus, establishing that RP = coRP or BPP = P would require proving superpolynomial lower bounds for Boolean or arithmetic circuits. We also show that any derandomization of RNC would yield new circuit lower bounds for a language in NEXP.
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-wisewhere each x i ∈ {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 , there is an efficient randomized algorithm A with the following property. Given a circuit C that computes f k on at least fraction of inputs, the algorithm A outputs with probability at least 3/4 a list of O(1/ ) 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/ )/k); moreover, each output circuit is an AC 0 circuit (of size poly(n, k, log 1/δ, 1/ )), with oracle access to the circuit C. Using the Goldreich-Levin decoding algorithm [GL89], we also get a fully uniform version of Yao's XOR Lemma [Yao82] with optimal parameters, up to constant factors. Our results simplify and improve those in [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 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.
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