Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds
 Valentine Kabanets,
 Russell Impagliazzo
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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 polynomialsize arithmetic circuits. We also prove a (partial) converse: If Permanent requires superpolynomialsize 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 polynomialsize 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 \({\text{NEXP}} \not\subset {\text{P/poly}}\) if both BPP = P and lowdegree testing is in P; here lowdegree testing is the problem of checking whether a given Boolean circuit computes a function that is close to some lowdegree polynomial over a finite field.
 Title
 Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds
 Journal

computational complexity
Volume 13, Issue 12 , pp 146
 Cover Date
 200412
 DOI
 10.1007/s0003700401826
 Print ISSN
 10163328
 Online ISSN
 14208954
 Publisher
 BirkhĂ¤userVerlag
 Additional Links
 Topics
 Keywords

 circuit lower bounds
 derandomization
 polynomial identity testing
 hardnessrandomness tradeoffs
 68Q10
 68Q15
 68Q17
 Industry Sectors
 Authors

 Valentine Kabanets ^{(1)}
 Russell Impagliazzo ^{(2)}
 Author Affiliations

 1. School of Computing Science, Simon Fraser University, Vancouver, BC V5A 1S6, Canada
 2. Department of Computer Science, University of California, San Diego, La Jolla, CA, 920930114, U.S.A