Revised: July 26, 2018

Published: December 18, 2018

**Keywords:**algebraic circuits, lower bounds, derandomization, Polynomial Identity Testing, barriers

**Categories:**complexity theory, lower bounds, arithmetic circuits, derandomization, polynomials - multivariate, Polynomial Identity Testing

**ACM Classification:**F.1.3

**AMS Classification:**68Q17, 52C07, 11H06, 11H31, 05B40

**Abstract:**
[Plain Text Version]

We formalize a framework of *algebraically natural* lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich (1997) for Boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike
in
the Boolean setting, there has been no concrete evidence demonstrating that this is a *barrier* to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding
whether algebraic circuits are expressive enough to support “cryptography” secure against algebraic circuits.

Following a similar result of Williams (2016) in the Boolean setting, we show that the existence of an algebraic natural proofs barrier is *equivalent* to the existence of *succinct* derandomization of the polynomial identity testing problem,
that is, to the existence of a hitting set for the class of $\poly(N)$-degree $\poly(N)$-size circuits which consists of coefficient vectors of polynomials of
$\polylog(N)$ degree
with $\polylog(N)$-size circuits.
Further, we give an explicit universal construction showing that *if* such a succinct hitting set exists, then our universal construction suffices.

Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that *none* of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier.

Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni (2001), except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.

A conference version of this paper appeared in the Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC 2017).