Embedding cover-free families and cryptographical applications

Cover-free families are set systems used as solutions for a large variety of problems, and in particular, problems where we deal with $n$ elements and want to identify $d$ invalid ones among them by performing only $t$ tests ($t \leq n$). We are specially interested in cryptographic problems, and we note that some of these problems need cover-free families with an increasing size $n$. Solutions that propose the increase of $n$, such as \emph{monotone families} and \emph{nested families}, have been recently considered in the literature. In this paper, we propose a generalization that we call \emph{embedding families}, which allows us to increase both $n$ and $d$. We propose constructions of \emph{embedding families} using polynomials over finite fields, and show specific cases where this construction allows us to prioritize increase of $d$ or $n$ with good compression ratios. We also provide new constructions for monotone families with improved compression ratio. Finally, we show how to use embedded sequences of orthogonal arrays and packing arrays to build embedding families.