Abstract
Let \({\mathcal {F}}\) be a set of functions of \(f : X \rightarrow Y\), where \(|X|=k, \,|Y|=v\) and \(|{\mathcal {F}}|=N\). If for any \(t\)-subset \(C \subseteq X\) there exists at least one function \(f\in \mathcal {F}\) such that \(f|_{C}\) is one-to-one, then \({\mathcal {F}}\) is called a perfect hash family, denoted by PHF\((N; k, v, t)\). In this paper, we construct the simplest nontrivial PHFs of \(t=3\) and \(N=3\) using classic generalized quadrangles, quadrics in PG\((4, q)\) and Hermitian varieties in PG\((4, q^2)\). We obtained PHF\((3; q^2(q+1), q^2, 3)\) and PHF\((3; q^5, q^3, 3)\) for \(q\) a prime power. The curve \(k= v^{5/3}\) is greater than known \(k\) for \(v=q^3,\,q\) a prime power.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Cryptography, Codes, Designs and Finite Fields: In Memory of Scott A. Vanstone”.
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Fuji-Hara, R. Perfect hash families of strength three with three rows from varieties on finite projective geometries. Des. Codes Cryptogr. 77, 351–356 (2015). https://doi.org/10.1007/s10623-015-0052-z
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DOI: https://doi.org/10.1007/s10623-015-0052-z