There are no post-quantum weakly pseudo-free families in any nontrivial variety of expanded groups

Let Formula: see text be a finite set of finitary operation symbols and let Formula: see text be a nontrivial variety of Formula: see text-algebras. Assume that for some set Formula: see text of group operation symbols, all Formula: see text-algebras in Formula: see text are groups under the operations associated with the symbols in Formula: see text. In other words, Formula: see text is assumed to be a nontrivial variety of expanded groups. In particular, Formula: see text can be a nontrivial variety of groups or rings. Our main result is that there are no post-quantum weakly pseudo-free families in Formula: see text, even in the worst-case setting and/or the black-box model. In this paper, we restrict ourselves to families Formula: see text of computational and black-box Formula: see text-algebras (where Formula: see text) such that for every Formula: see text, each element of Formula: see text 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 Formula: see text-algebras. Note that under certain conditions, this result depends on the classification of finite simple groups. Also, we define and study some types of post-quantum weak pseudo-freeness for families of computational and black-box Formula: see text-algebras.