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

Mikhail Anokhin

TL;DR

This work addresses whether post-quantum weakly pseudo-free families exist within nontrivial varieties of expanded groups. It develops a universal-algebraic framework for computational and black-box Ω-algebras, and introduces several variants of weak pseudo-freeness, including post-quantum and worst-case versions. The authors prove a strong negative result: there are no post-quantum weakly pseudo-free families in any nontrivial expanded-group variety, even under black-box and worst-case assumptions, by reducing to black-box groups in a subvariety 𝔙|_Γ and constructing quantum algorithms based on order-finding and constructive-membership that break potential candidates. The result hinges on reductions to Ω-reducts, straight-line-program representations of relations, and (where needed) the Classification of Finite Simple Groups, and it has implications for cryptographic constructions based on algebraic structures, suggesting a shift toward non-expanded structures or alternate notions of pseudo-freeness for post-quantum security analyses.

Abstract

Let $Ω$ be a finite set of finitary operation symbols and let $\mathfrak V$ be a nontrivial variety of $Ω$-algebras. Assume that for some set $Γ\subseteqΩ$ of group operation symbols, all $Ω$-algebras in $\mathfrak V$ are groups under the operations associated with the symbols in $Γ$. 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 $Ω$-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 $d$. In our main result, we use straight-line programs to represent nontrivial relations between elements of $Ω$-algebras. 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.

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

TL;DR

This work addresses whether post-quantum weakly pseudo-free families exist within nontrivial varieties of expanded groups. It develops a universal-algebraic framework for computational and black-box Ω-algebras, and introduces several variants of weak pseudo-freeness, including post-quantum and worst-case versions. The authors prove a strong negative result: there are no post-quantum weakly pseudo-free families in any nontrivial expanded-group variety, even under black-box and worst-case assumptions, by reducing to black-box groups in a subvariety 𝔙|_Γ and constructing quantum algorithms based on order-finding and constructive-membership that break potential candidates. The result hinges on reductions to Ω-reducts, straight-line-program representations of relations, and (where needed) the Classification of Finite Simple Groups, and it has implications for cryptographic constructions based on algebraic structures, suggesting a shift toward non-expanded structures or alternate notions of pseudo-freeness for post-quantum security analyses.

Abstract

Let be a finite set of finitary operation symbols and let be a nontrivial variety of -algebras. Assume that for some set of group operation symbols, all -algebras in are groups under the operations associated with the symbols in . In other words, is assumed to be a nontrivial variety of expanded groups. In particular, can be a nontrivial variety of groups or rings. Our main result is that there are no post-quantum weakly pseudo-free families in , even in the worst-case setting and/or the black-box model. In this paper, we restrict ourselves to families of computational and black-box -algebras (where ) such that for every , each element of is represented by a unique bit string of length polynomial in the length of . In our main result, we use straight-line programs to represent nontrivial relations between elements of -algebras. 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 -algebras.
Paper Structure (21 sections, 7 theorems, 25 equations, 1 figure)

This paper contains 21 sections, 7 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution and Organization of the Paper
  4. Preliminaries
  5. General Preliminaries
  6. Universal-Algebraic Preliminaries
  7. Group-Theoretic Preliminaries
  8. Probabilistic Preliminaries
  9. Cryptographic Preliminaries
  10. Weakly Pseudo-Free Families of Computational and Black-Box -Algebras
  11. Standard Model
  12. Black-Box -Algebra Model
  13. Quantum Computation Model
  14. Relations between the Types of Weak Pseudo-Freeness
  15. Weak Pseudo-Freeness in the Variety Generated by the -Reducts of All -Algebras in V
  16. ...and 6 more sections

Key Result

Lemma 2.1

Suppose $F$ is an $\Omega$-algebra in $\mathfrak V$ and $(f_i\,|\, i\in I)$ is a generating system of $F$. Then $F$ is a $\mathfrak V$-free $\Omega$-algebra freely generated by $(f_i\,|\, i\in I)$ if and only if for any $m\in\mathbb N$ and any $v,w\in T_m$, the identity $v=w$ holds in $\mathfrak V$

Figures (1)

  • Figure 1: Relations between the types of weak pseudo-freeness defined in Section \ref{['s:wpsfreefams']}. The abbreviations PQ, WC, FoC$\Omega$A, FoBB$\Omega$A, and w/oD stand for Post-Quantum, Worst-Case, (weakly pseudo-free) Family of Computational $\Omega$-Algebras, (weakly pseudo-free) Family of Black-Box $\Omega$-Algebras, and without Distributions, respectively. For brevity, we do not write an abbreviation for Weakly Pseudo-Free in the diagram. Weak pseudo-freeness of any type means weak pseudo-freeness of this type in $\mathfrak V$ with respect to $(\mathcal{D}_k\,|\, k\in K)$ (or $(D_k\,|\, k\in K)$ in the worst-case setting) and $\sigma$. A horizontal double-line arrow from $Y$ to $Z$ labeled "$\text{if }\forall\,k\ (\mathop{\mathrm{supp}}\nolimits\mathcal{D}_k\subseteq D_k)$" means that if $((H_d,\mathcal{H}_d)\,|\, d\in D)$ is a family of type $Y$ and $\mathop{\mathrm{supp}}\nolimits\mathcal{D}_k\subseteq D_k$ for all $k\in K$, then $(H_d\,|\, d\in D)$ is a family of type $Z$ (see Remark \ref{['r:wpsfreeiswcwpsfree']}). Furthermore, a vertical unlabeled double-line arrow from $Y$ to $Z$ means that any family of type $Y$ is also a family of type $Z$ (see Remark \ref{['r:pqwpsfreeiswpsfree']}). Finally, an oblique double-line arrow from $Y$ to $Z$ labeled "$\exists$" means that if there exists a family of type $Y$, then there exists a family of type $Z$ (see Proposition \ref{['p:exwpffocOaimplexwpffobbOa']}).

Theorems & Definitions (30)

  • Lemma 2.1
  • Lemma 2.2
  • Example 3.1: representation of elements of $F_\infty(\mathfrak V)$ by straight-line programs, see also Ano21 or Ano22
  • Definition 3.2: family of computational $\Omega$-algebras without distributions
  • Definition 3.3: family of computational $\Omega$-algebras (with distributions), see also Ano21 or Ano22
  • Definition 3.4: family is in $\mathfrak V$
  • Definition 3.5: weakly pseudo-free family of computational $\Omega$-algebras
  • Definition 3.7: worst-case weakly pseudo-free family of computational $\Omega$-algebras without distributions
  • Definition 3.8: black-box $\Omega$-algebra
  • Definition 3.9: $\Omega$-oracle
  • ...and 20 more