  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.

Cryptography and Security;Computational Complexity