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

Abstract

Let $\mathrm{\Omega }$ be a finite set of finitary operation symbols and let $𝔙$ be a nontrivial variety of $\mathrm{\Omega }$-algebras. Assume that for some set $\mathrm{\Gamma }\subseteq \mathrm{\Omega }$ of group operation symbols, all $\mathrm{\Omega }$-algebras in $𝔙$ are groups under the operations associated with the symbols in $\mathrm{\Gamma }$. In other words, $𝔙$ is assumed to be a nontrivial variety of expanded groups. In particular, $𝔙$ can be a nontrivial variety of groups or rings. Our main result is that there are no post-quantum weakly pseudo-free families in $𝔙$, even in the worst-case setting and/or the black-box model. In this paper, we restrict ourselves to families $(H_d\mathrm{|d}\in D)$ of computational and black-box $\mathrm{\Omega }$-algebras (where $D\subseteq {\{0,1\}}^{\ast }$) 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 $d$. In our main result, we use straight-line programs to represent nontrivial relations between elements of $\mathrm{\Omega }$-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 $\mathrm{\Omega }$-algebras.