In a Nisan-Wigderson design polynomial (in short, a design polynomial), the gcd of every pair of monomials has a low degree. A useful example of such a polynomial is the following: NW d k ( x ) = h F d [ z ] deg ( h ) k d − 1 i =0 x i h ( i ) 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, let us fix k = d . The family of polynomials : = { NW d k : d is a prime } and close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit lower bound proofs. But, unlike the permanent, very little is known about the various complexity and structural aspects of beyond the fact that VNP . Is VNP -complete? Is NW d k characterized by its symmetries? Is it circuit-testable, i.e., given a circuit C can we check efficiently if C computes NW d k ? Characterization of polynomials by their symmetries plays a central role in the geometric complexity theory program. Here, we answer the last two questions.

We show that NW d k is characterized by its group of symmetries over C . 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 implication, we obtain the ``flip theorem'' for : Suppose NW d k is not computable by circuits of size s . Then, there exist poly ( s ) many points a 1 a m such that for every circuit C of size s , there is an [ m ] satisfying C ( a ) = NW d k ( a ) . These points can be computed deterministically in poly ( s ) time if black-box polynomial identity testing for size- s circuits can be derandomized in poly ( s ) time. It is well-known that the permanent polynomial has the above-mentioned features.

We also show a structural similarity between the group of symmetries of the permanent and that of NW d k : 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.