Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters

TL;DR

This paper presents the first explicit, deterministic algorithm for constructing near-optimal orthogonal arrays of arbitrary parameters, using algebraic geometry codes, with applications to universal hashing.

Contribution

It provides the first explicit, algorithmic construction of orthogonal arrays for all parameters, extending to efficient $t$-independent hash functions for arbitrary domains and codomains.

Findings

Constructed orthogonal arrays with near-optimal size for all parameters.

Developed efficient $t$-independent hash functions for arbitrary domain and codomain.

Used algebraic geometry codes in the construction.

Abstract

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.