 
Summary: Algebrization: A New Barrier in Complexity Theory
Scott Aaronson
MIT
Avi Wigderson
Institute for Advanced Study
Abstract
Any proof of P = NP will have to overcome two barriers: relativization and natural proofs.
Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not
have linearsize circuits) that overcome both barriers simultaneously. So the question arises of
whether there is a third barrier to progress on the central questions in complexity theory.
In this paper we present such a barrier, which we call algebraic relativization or algebriza
tion. The idea is that, when we relativize some complexity class inclusion, we should give the
simulating machine access not only to an oracle A, but also to a lowdegree extension of A over
a finite field or ring.
We systematically go through basic results and open problems in complexity theory to delin
eate the power of the new algebrization barrier. First, we show that all known nonrelativizing
results based on arithmetizationboth inclusions such as IP = PSPACE and MIP = NEXP, and
separations such as MAEXP P/poly do indeed algebrize. Second, we show that almost all of
the major open problemsincluding P versus NP, P versus RP, and NEXP versus P/polywill
require nonalgebrizing techniques. In some cases algebrization seems to explain exactly why
