An $n$-variate polynomial $g$ of degree $d$ is a $(n,d,t)$ design polynomial if the degree of the gcd of every pair of monomials of $g$ is at most $t-1$. The power symmetric polynomial $\mathrm{PSym}_{n,d} := \sum_{i=1}^{n} x^d_i$ and the sum-product polynomial $\mathrm{SP}_{s,d} := \sum_{i=1}^{s}\prod_{j=1}^{d} x_{i,j}$ are instances of design polynomials for $t=1$. Another example is the Nisan-Wigderson design polynomial $\mathrm{NW}$, which has been used extensively to prove various arithmetic circuit lower bounds. Given black-box access to an $n$-variate, degree-$d$ polynomial $f(\mathbf{x}) \in \mathbb{F}[\mathbf{x}]$, how fast can we check if there exist an $A \in \mathrm{GL}(n, \mathbb{F})$ and a $\mathbf{b} \in \mathbb{F}^n$ such that $f(A\mathbf{x}+\mathbf{b})$ is a $(n,d,t)$ design polynomial? We call this problem "testing equivalence to design polynomials", or alternatively, "equivalence testing for design polynomials".

In this work, we present a randomized algorithm that finds $(A, \mathbf{b})$ such that $f(A\mathbf{x}+\mathbf{b})$ is a $(n,d,t)$ design polynomial, if such $A$ and $\mathbf{b}$ exist, provided $t \leq d/3$. The algorithm runs in $(nd)^{O(t)}$ time and works over any sufficiently large $\mathbb{F}$ of characteristic $0$ or $> d$. As applications of this test, we show two results -- one is structural and the other is algorithmic. The structural result establishes a polynomial-time equivalence between the graph isomorphism problem and the polynomial equivalence problem for design polynomials. The algorithmic result implies that Patarin's scheme [Patarin,EUROCRYPT 1996] can be broken in quasi-polynomial time if a random sparse polynomial is used in the key generation phase.

We also give an efficient learning algorithm for $n$-variate random affine projections of multilinear degree-$d$ design polynomials, provided $n \geq d^4$. If one obtains an analogous result under the weaker assumption "$n \geq d^{\epsilon}$, for any $\epsilon > 0$", then the $\mathrm{NW}$ family is not $\mathrm{VNP}$-complete unless there is a $\mathrm{VNP}$-complete family whose random affine projections are learnable. It is not known if random affine projections of the permanent are learnable.

The above algorithms are obtained by using the vector space decomposition framework, introduced in [Kayal-Saha, STOC 2019] and [Garg-Kayal-Saha, FOCS 2020], for learning non-degenerate arithmetic circuits. A key technical difference between the analysis in [Garg-Kayal-Saha, FOCS 2020] and in [Bhargava-Garg-Kayal-Saha, RANDOM 2022] and the analysis here is that a certain adjoint algebra, which turned out to be trivial (i.e., diagonalizable) in prior works, is non-trivial in our case. However, we show that the adjoint arising here is triangularizable which then helps in carrying out the vector space decomposition step.