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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.
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.
In this article, we study the connections between pseudo-free families of computational Ω \Omega -algebras (in appropriate varieties of Ω \Omega -algebras for suitable finite sets Ω \Omega of finitary operation symbols) and certain standard cryptographic primitives. We restrict ourselves to families ( H d ∣ d ∈ D ) \left({H}_{d}\hspace{0.33em}| \hspace{0.33em}d\in D) of computational Ω \Omega -algebras (where D ⊆ { 0 , 1 } ∗ D\subseteq {\left\{0,1\right\}}^{\ast } ) such that for every d ∈ D d\in D , each element of H d {H}_{d} is represented by a unique bit string of the length polynomial in the length of d d . Very loosely speaking, our main results are as follows: (i) pseudo-free families of computational mono-unary algebras with one to one fundamental operation (in the variety of all mono-unary algebras) exist if and only if one-way families of permutations exist; (ii) for any m ≥ 2 m\ge 2 , pseudo-free families of computational m m -unary algebras with one to one fundamental operations (in the variety of all m m -unary algebras) exist if and only if claw resistant families of m m -tuples of permutations exist; (iii) for a certain Ω \Omega and a certain variety V {\mathfrak{V}} of Ω \Omega -algebras, the existence of pseudo-free families of computational Ω \Omega -algebras in V {\mathfrak{V}} implies the existence of families of trapdoor permutations.
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