Towards a tight hardness-randomness connection between permanent and arithmetic circuit identity testing

In this paper we make progress on establishing a tight connection between the problem of derandomization of arithmetic circuit identity testing (ACIT), and the arithmetic circuit complexity of the permanent defined by pern = ∑ σ∈Sn ∏n i=1 xiσ(i). We develop an ACIT-based derandomization hypothesis, and show this is a necessary condition for proving that permanent has super-polynomial arithmetic circuits over F, for fields F of characteristic zero. Informally, this hypothesis poses the existence of a subexponential size hitting set Hn computable by subexponential size uniform TC circuits against size n arithmetic circuits with m ≤ n variables whose output is multilinear. Assuming the Generalized Riemann Hypothesis (GRH), it can be shown that this hypothesis is sufficient for showing that either permanent does not have polynomial size (nonuniform) arithmetic circuits, or that the Boolean circuit class uniform TC is strictly contained in uniform NC. Without (GRH), the hypothesis implies such a disjunction, but with the first item stating permanent does not have polynomial size constant-free arithmetic circuits. In this setting the converse also goes through, but based on the slightly stronger assumption that all constant multiples an · pern require super-polynomial constant-free arithmetic circuits, for an ∈ Z/{0} computable by poly(n) size constant-free circuits.