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

02/21/2023

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by   Mikhail Anokhin, et al.

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Let Ω be a finite set of finitary operation symbols and let 𝔙 be a nontrivial variety of Ω-algebras. Assume that for some set Γ⊆Ω of group operation symbols, all Ω-algebras in 𝔙 are groups under the operations associated with the symbols in Γ. 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|d∈ D) of computational and black-box Ω-algebras (where D⊆{0,1}^*) such that for every d∈ 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 Ω-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 Ω-algebras.