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 $d$. We use straight-line programs to represent nontrivial relations between elements of $\Omega$-algebras in our main result. 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：设$\Omega$为有限个有限元运算符号集，$\mathfrak V$为$\Omega-代数的一个非平凡簇。假设存在某个群运算符号子集$\Gamma\subseteq\Omega$，使得$\mathfrak V$中所有$\Omega-代数在$\Gamma$中符号对应的运算下构成群。换言之，$\mathfrak V$被假定为非平凡扩张群簇。特别地，$\mathfrak V$可以是群或环的非平凡簇。我们的主要结论是：$\mathfrak V$中不存在后量子弱伪自由族，即使在最坏情形设置和/或黑盒模型下也是如此。本文限制考虑计算型与黑盒型$\Omega-代数族$(H_d\mathbin|d\in D)$（其中$D\subseteq\{0,1\}^*$）满足：对任意$d\in D$，$H_d$中每个元素均由长度关于$d$长度呈多项式增长的唯一比特串表示。我们利用直线程序表述主要结论中$\Omega-代数元素间的非平凡关系。需注意，该结论在特定条件下依赖于有限单群分类定理。此外，我们定义并研究了计算型与黑盒型$\Omega-代数族的若干弱伪自由性变体。