We study the problems of testing isomorphism of 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 $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.
翻译:我们研究的是多面形、代数和多线形的测试问题。 我们的第一种主要结果就是这些问题的平均量子算法。 例如, 我们开发了一种算法, 以两种立方形式$f, g\in\mathbb{F ⁇ q[x_1,\dots, x_n]$, 并且决定对于大部分美元来说, 美元和 美元是否是时间的异形 $q ⁇ O(n)}。 这种平均情况设置具有直接的实际影响, 自1990年代以来, 一直在多变式加密学中研究过。 我们的第二个结果涉及交替三线形体的测试等同的复杂性。 这个问题在数学和加密学中都引起兴趣。 我们通过显示它们既是Tensor Isoforismismism-complement( Grochow- Qiao, ITCS, 2021), 因而相当于对大多数领域立体形态的测试。