Pseudo-free families and cryptographic primitives

Abstract

Abstract 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.