**Pseudo-Free Families of Computational Universal Algebras**

*Mikhail Anokhin*

**Abstract: **Let $\Omega$ be a finite set of operation symbols. We initiate the study of (weakly) pseudo-free families of computational $\Omega$-algebras in arbitrary varieties of $\Omega$-algebras. Most of our results concern (weak) pseudo-freeness in the variety $\mathfrak O$ of all $\Omega$-algebras. A family $(H_d)_{d\in D}$ of computational $\Omega$-algebras (where $D\subseteq\{0,1\}^*$) is called polynomially bounded (resp., having exponential size) if there exists a polynomial $\eta$ such that for all $d\in D$, the length of any representation of every $h\in H_d$ is at most $\eta(\lvert d\rvert)$ (resp., $\lvert H_d\rvert\le2^{\eta(\lvert d\rvert)}$). First, we prove the following trichotomy: (i) if $\Omega$ consists of nullary operation symbols only, then there exists a polynomially bounded pseudo-free family in $\mathfrak O$; (ii) if $\Omega=\Omega_0\cup\{\omega\}$, where $\Omega_0$ consists of nullary operation symbols and the arity of $\omega$ is $1$, then there exist an exponential-size pseudo-free family and a polynomially bounded weakly pseudo-free family (both in $\mathfrak O$); (iii) in all other cases, the existence of polynomially bounded weakly pseudo-free families in $\mathfrak O$ implies the existence of collision-resistant families of hash functions. Second, assuming the existence of collision-resistant families of hash functions, we construct a polynomially bounded weakly pseudo-free family and an exponential-size pseudo-free family of computational $m$-ary groupoids (both in $\mathfrak O$), where $m$ is an arbitrary positive integer. In particular, for arbitrary $m\ge2$, polynomially bounded weakly pseudo-free families of computational $m$-ary groupoids in $\mathfrak O$ exist if and only if collision-resistant families of hash functions exist. Moreover, we present some simple constructions of cryptographic primitives from pseudo-free families satisfying certain additional conditions. These constructions demonstrate the potential of pseudo-free families.

**Category / Keywords: **foundations / universal algebras, families of computational universal algebras, pseudo-freeness, weak pseudo-freeness, hash functions, collision resistance, $n$-ary groupoids

**Date: **received 2 Dec 2018, last revised 4 Dec 2018

**Contact author: **anokhin at mccme ru

**Available format(s): **PDF | BibTeX Citation

**Version: **20181204:140746 (All versions of this report)

**Short URL: **ia.cr/2018/1178

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