## Derandomizing Polynomial Identity Tests means proving circuit lower bounds

### Abstract

We show that derandomizing the 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 the Polynomial Identity Testing is a coRP problem, we obtain the following corollary: If RP=P (or, even, coRP ⊆ ∩ε>0 NTIME(2nε), 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.

Our techniques allow us to prove an unconditional circuit lower bound for a language in NEXPRP: we prove that either (i) Permanent is not computable by polynomial-size arithmetic circuits, or (ii) NEXPRP ⊄ P/poly.

Finally, we prove that NEXP ⊄ P/poly if both BPP=P and the low-degree testing is in P; here, the 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.

### Versions

• journal version in Computational Complexity, 13(1-2), pages 1-46, 2004.
• extended abstract in Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing (STOC'03), pages 355-364, 2003.