Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929343515590656 |
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| author | Harvey, Nicholas Sahami, Arvin |
| author_facet | Harvey, Nicholas Sahami, Arvin |
| contents | Orthogonal arrays are a type of combinatorial design that were developed in the 1940s in the design of statistical experiments. In 1947, Rao proved a lower bound on the size of any orthogonal array, and raised the problem of constructing arrays of minimum size. Kuperberg, Lovett and Peled (2017) gave a non-constructive existence proof of orthogonal arrays whose size is near-optimal (i.e., within a polynomial of Rao's lower bound), leaving open the question of an algorithmic construction. We give the first explicit, deterministic, algorithmic construction of orthogonal arrays achieving near-optimal size for all parameters. Our construction uses algebraic geometry codes.
In pseudorandomness, the notions of $t$-independent generators or $t$-independent hash functions are equivalent to orthogonal arrays. Classical constructions of $t$-independent hash functions are known when the size of the codomain is a prime power, but very few constructions are known for an arbitrary codomain. Our construction yields algorithmically efficient $t$-independent hash functions for arbitrary domain and codomain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08787 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters Harvey, Nicholas Sahami, Arvin Data Structures and Algorithms Computational Complexity Combinatorics Statistics Theory Orthogonal arrays are a type of combinatorial design that were developed in the 1940s in the design of statistical experiments. In 1947, Rao proved a lower bound on the size of any orthogonal array, and raised the problem of constructing arrays of minimum size. Kuperberg, Lovett and Peled (2017) gave a non-constructive existence proof of orthogonal arrays whose size is near-optimal (i.e., within a polynomial of Rao's lower bound), leaving open the question of an algorithmic construction. We give the first explicit, deterministic, algorithmic construction of orthogonal arrays achieving near-optimal size for all parameters. Our construction uses algebraic geometry codes. In pseudorandomness, the notions of $t$-independent generators or $t$-independent hash functions are equivalent to orthogonal arrays. Classical constructions of $t$-independent hash functions are known when the size of the codomain is a prime power, but very few constructions are known for an arbitrary codomain. Our construction yields algorithmically efficient $t$-independent hash functions for arbitrary domain and codomain. |
| title | Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters |
| topic | Data Structures and Algorithms Computational Complexity Combinatorics Statistics Theory |
| url | https://arxiv.org/abs/2405.08787 |