Pseudo-free families of computational universal algebras
| Autor: | M I Anokhin |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
08a62
08a70 Computer science Applied Mathematics 020206 networking & telecommunications n-ary groupoid 02 engineering and technology 68q17 collision-resistant family of hash functions Computer Science Applications Computational Mathematics pseudo-free family universal algebra QA1-939 0202 electrical engineering electronic engineering information engineering 020201 artificial intelligence & image processing weakly pseudo-free family 94a60 Mathematics |
| Zdroj: | Journal of Mathematical Cryptology, Vol 15, Iss 1, Pp 197-222 (2020) |
| ISSN: | 1862-2984 |
| DOI: | 10.1515/jmc-2020-0014 |
| Popis: | 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 ., | H d | ≤ 2 η ( | d | ) ) . $\eta (|d|)\left( \text{ resp}\text{., }\left| {{H}_{d}} \right|\le {{2}^{\eta (|d|)}} \right).$ First, we prove the following trichotomy: (i) if Ω consists of nullary operation symbols only, then there exists a polynomially bounded pseudo-free family; (ii) if Ω = Ω 0 ∪ {ω}, where Ω 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. |
| Databáze: | OpenAIRE |
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