Pseudo-free families of computational universal algebras

Abstract

Let ω be a finite set of finitary operation symbols. We initiate the study of (weakly) pseudo-free families of computational ω-algebras in arbitrary varieties of ω-algebras. A family (Hd | d ∈ D) of computational ω-algebras (where D ⊆ {0, 1}*) is called polynomially bounded (resp., having exponential size) if there exists a polynomial η such that for all d ∈ D, the length of any representation of every h ∈ Hd is at most η(|d|)(resp., | Hd |≤2η(|d|)). First, we prove the following trichotomy: (i) if consists of nullary operation symbols only, then there exists a polynomially bounded pseudo-free family; (ii) if ω = ω0 ⊂ {ω}, where 0 consists of nullary operation symbols and the arity of ω is 1, then there exist an exponential-size pseudo-free family and a polynomially bounded weakly pseudo-free family; (iii) in all other cases, the existence of polynomially bounded weakly pseudo-free families implies the existence of collision-resistant families of hash functions. In this trichotomy, (weak) pseudo-freeness is meant in the variety of all ω-algebras. 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 in the variety of all m-ary groupoids, where m is an arbitrary positive integer.