Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds

Abstract.

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) ${\text{NEXP}} \not\subset {\text{P}}/{\text{poly or}}$ (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 ${\text{RP = P}}{\kern 1pt} {\text{(or even coRP }} \subseteq \cap _{\varepsilon > 0} {\text{ NTIME}}(2^{n^\varepsilon } ),{\text{ infinitely often),}}$ 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 polynomial-size Boolean or arithmetic circuits. Finally, we show that ${\text{NEXP}} \not\subset {\text{P/poly}}$ 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.