On an almost-universal hash function family with applications to authentication and secrecy codes

Universal hashing, discovered by Carter and Wegman in 1979, has many important applications in computer science. MMH${}^{\ast }$, which was shown to be $\mathrm{\Delta }$-universal by Halevi and Krawczyk in 1997, is a well-known universal hash function family. We introduce a variant of MMH${}^{\ast }$, that we call GRDH, where we use an arbitrary integer $n>1$ instead of prime $p$ and let the keys $\mathbf{x}=⟨x_1,\dots ,x_k⟩\in {\mathbb{Z}}_n^k$ satisfy the conditions $\gcd (x_i,n)=t_i$ ($1\le i\le k$), where $t_1,\dots ,t_k$ are given positive divisors of $n$. Then via connecting the universal hashing problem to the number of solutions of restricted linear congruences, we prove that the family GRDH is an $\epsilon$-almost-$\mathrm{\Delta }$-universal family of hash functions for some $\epsilon <1$ if and only if $n$ is odd and $\gcd (x_i,n)=t_i=1$ $(1\le i\le k)$. Furthermore, if these conditions are satisfied then GRDH is $\frac{1}{p-1}$-almost-$\mathrm{\Delta }$-universal, where $p$ is the smallest prime divisor of $n$. Finally, as an application of our results, we propose an authentication code with secrecy scheme which strongly generalizes the scheme studied by Alomair et al. [{\it J. Math. Cryptol.} {\bf 4} (2010), 121–148], and [{\it J.UCS} {\bf 15} (2009), 2937–2956].