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• W3115431341 abstract "Abstract In this article, we study the connections between pseudo-free families of computational <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>Ω</m:mi> </m:math> Omega -algebras (in appropriate varieties of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>Ω</m:mi> </m:math> Omega -algebras for suitable finite sets <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>Ω</m:mi> </m:math> Omega of finitary operation symbols) and certain standard cryptographic primitives. We restrict ourselves to families <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msub> <m:mspace width=0.33em /> <m:mo>∣</m:mo> <m:mspace width=0.33em /> <m:mi>d</m:mi> <m:mo>∈</m:mo> <m:mi>D</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> left({H}_{d}hspace{0.33em}| hspace{0.33em}din D) of computational <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>Ω</m:mi> </m:math> Omega -algebras (where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>D</m:mi> <m:mo>⊆</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> </m:math> Dsubseteq {left{0,1right}}^{ast } ) such that for every <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>d</m:mi> <m:mo>∈</m:mo> <m:mi>D</m:mi> </m:math> din D , each element of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msub> </m:math> {H}_{d} is represented by a unique bit string of the length polynomial in the length of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>d</m:mi> </m:math> 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:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> mge 2 , pseudo-free families of computational <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>m</m:mi> </m:math> m -unary algebras with one to one fundamental operations (in the variety of all <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>m</m:mi> </m:math> m -unary algebras) exist if and only if claw resistant families of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>m</m:mi> </m:math> m -tuples of permutations exist; (iii) for a certain <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>Ω</m:mi> </m:math> Omega and a certain variety <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=fraktur>V</m:mi> </m:math> {mathfrak{V}} of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>Ω</m:mi> </m:math> Omega -algebras, the existence of pseudo-free families of computational <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>Ω</m:mi> </m:math> Omega -algebras in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=fraktur>V</m:mi> </m:math> {mathfrak{V}} implies the existence of families of trapdoor permutations." @default.
• W3115431341 created "2021-01-05" @default.
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• W3115431341 date "2022-01-01" @default.
• W3115431341 modified "2023-09-25" @default.
• W3115431341 title "Pseudo-free families and cryptographic primitives" @default.
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• W3115431341 doi "https://doi.org/10.1515/jmc-2020-0055" @default.
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