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

Abstract

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 [Formula: see text]. 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.