### On the Symmetries of and Equivalence Test for Design Polynomials

Nikhil Gupta, Chandan Saha, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science
In a Nisan-Wigderson design polynomial (in short, a design polynomial), every pair of monomials share a few common variables. A useful example of such a polynomial, introduced in [34], is the following: where d is a prime, F d is the finite field with d elements, and k d. The degree of the gcd of every pair of monomials in NW d,k is at most k. For concreteness, we fix k = √ d . The family of polynomials N W := {NW d,k : d is a prime} and close variants of it have been used as hard explicit
more » ... omial families in several recent arithmetic circuit lower bound proofs. But, unlike the permanent, very little is known about the various structural and algorithmic/complexity aspects of N W beyond the fact that N W ∈ VNP. Is NW d,k characterized by its symmetries? Is it circuittestable, i.e., given a circuit C can we check efficiently if C computes NW d,k ? What is the complexity of equivalence test for N W, i.e., given black-box access to a f ∈ F[x], can we check efficiently if there exists an invertible linear transformation A such that f = NW d,k (A • x)? Characterization of polynomials by their symmetries plays a central role in the geometric complexity theory program. Here, we answer the first two questions and partially answer the third. We show that NW d,k is characterized by its group of symmetries over C, but not over R. We also show that NW d,k is characterized by circuit identities which implies that NW d,k is circuit-testable in randomized polynomial time. As another application of this characterization, we obtain the "flip theorem" for N W. We give an efficient equivalence test for N W in the case where the transformation A is a block-diagonal permutation-scaling matrix. The design of this algorithm is facilitated by an almost complete understanding of the group of symmetries of NW d,k : We show that if A is in the group of symmetries of NW d,k then A = D • P , where D and P are diagonal and permutation matrices respectively. This is proved by completely characterizing the Lie algebra of NW d,k , and using an interplay between the Hessian of NW d,k and the evaluation dimension.