Let $\Omega$ be a finite set of finitary operation symbols and let $\mathfrak V$ be a nontrivial variety of $\Omega$-algebras. Assume that for some set $\Gamma\subseteq\Omega$ of group operation symbols, all $\Omega$-algebras in $\mathfrak V$ are groups under the operations associated with the symbols in $\Gamma$. In other words, $\mathfrak V$ is assumed to be a nontrivial variety of expanded groups. In particular, $\mathfrak V$ can be a nontrivial variety of groups or rings. Our main result is that there are no post-quantum weakly pseudo-free families in $\mathfrak V$, even in the worst-case setting and/or the black-box model. In this paper, we restrict ourselves to families $(H_d\mathbin|d\in D)$ of computational and black-box $\Omega$-algebras (where $D\subseteq\{0,1\}^*$) such that for every $d\in D$, each element of $H_d$ is represented by a unique bit string of length polynomial in the length of $d$. In our main result, we use straight-line programs to represent nontrivial relations between elements of $\Omega$-algebras. Note that under certain conditions, this result depends on the classification of finite simple groups. Also, we define and study some types of weak pseudo-freeness for families of computational and black-box $\Omega$-algebras.

翻译：令 $\Omega$ 为有限个有限元运算符号的集合，并设 $\mathfrak V$ 为一个非平凡的 $\Omega$-代数簇。假设对于某个群运算符号集 $\Gamma\subseteq\Omega$，$\mathfrak V$ 中的所有 $\Omega$-代数在 $\Gamma$ 中符号对应的运算下均构成群。换言之，$\mathfrak V$ 被假定为一个非平凡的扩张群簇。特别地，$\mathfrak V$ 可以是非平凡的群簇或环簇。我们的主要结论是：在 $\mathfrak V$ 中不存在后量子弱伪自由族，即使在最坏情况设定和/或黑盒模型中也是如此。在本文中，我们仅限于考虑计算型及黑盒型 $\Omega$-代数族 $(H_d\mathbin|d\in D)$（其中 $D\subseteq\{0,1\}^*$），使得对于每个 $d\in D$，$H_d$ 的每个元素均由长度与 $d$ 的长度成多项式关系的唯一比特串表示。在我们的主要结论中，我们使用直列程序来表示 $\Omega$-代数元素之间的非平凡关系。需注意，在特定条件下，该结论依赖于有限单群的分类。此外，我们定义并研究了计算型及黑盒型 $\Omega$-代数族的若干类弱伪自由性。