The Improvement of the Bound on Hash Family

TL;DR

This paper improves the theoretical bounds on three types of hash families using the Singleton bound, providing tighter limits under specific parameter conditions.

Contribution

It introduces new, tighter bounds for three classes of hash families, enhancing the theoretical understanding of their limitations.

Findings

New bounds are better for $\epsilon-U(N; n, m)$ hash families when $n>m^2>1$.

Improved bounds for $\epsilon-igtriangleup U(N; n, m)$ hash families when $n>m>1$.

Enhanced bounds for $\epsilon-SU(N; n, m)$ hash families when $n>2^m>2$.

Abstract

In this paper, we study the bound on three kinds of hash family using the Singleton bound. To $\epsilon-U(N; n, m)$ hash family, in the caes of $n>m^2>1$ and $1\geq\epsilon\geq \epsilon_1(n, m)$, we get that the new bound is better. To $\epsilon-\bigtriangleup U(N; n, m)$ hash family, in the case of $n>m>1$ and $1\geq\epsilon\geq\epsilon_3(n,m)$, the new bound is better. To $\epsilon-SU(N; n, m)$ hash family, in the case of $n>2^m>2$ and $1\geq\epsilon\geq \epsilon_4(n, m)$, we get that the new bound is better.