Average-case algorithms for testing isomorphism of polynomials, algebras, and multilinear forms

We study the problems of testing isomorphism polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, decides whether $f$ $g$ isomorphic in time $q^{O(n)}$ most $f$. This setting has direct practical implications, having been studied multivariate cryptography since 1990s. second result concerns complexity equivalence alternating trilinear problem is interest both mathematics cryptography. show this polynomial-time equivalent to symmetric forms, by showing they Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore over fields.