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Let $\Omega$ be a finite set of finitary operation symbols and let $\mathfrakV$ 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 varietyof expanded groups. In particular, $\mathfrak V$ can be a nontrivial variety ofgroups or rings. Our main result is that there are no post-quantum weaklypseudo-free families in $\mathfrak V$, even in the worst-case setting and/orthe 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 $H_d$ isrepresented by a unique bit string of length polynomial in the length of $d$.We use straight-line programs to represent nontrivial relations betweenelements of $\Omega$-algebras in our main result. Note that under certainconditions, this result depends on the classification of finite simple groups.Also, we define and study some types of weak pseudo-freeness for families ofcomputational and black-box $\Omega$-algebras.

There Are No Post-Quantum Weakly Pseudo-Free Families in Any Nontrivial Variety of Expanded Groups