Secure Sketch for Multi-Sets
Given the original set $X$ where $|X|=s$, a sketch $P$ is computed from $X$ and made public. From another set $Y$ where $|Y| = s$ and $P$, we can reconstruct $X$ if $|X\cap Y|\ge |s-t|$, where $t<s$ is some threshold. The sketch $P$ is secure if it does not reveal much information about $X$. A few constructions have been proposed, but they cannot handle multi-sets, that is, sets that may contain duplicate elements. We observe that the techniques in the set reconciliation protocol proposed by Minsky et al. (ISIT 2001) can be applied and give a secure sketch that supports multi-sets. If $X$ is a subset of an universe with $n$ elements, the running time of the encoding and decoding algorithms will be polynomial w.r.t. $s$ and $\log n$, and the entropy loss due to the sketch is less than $2t(1+\log n)$.
Small Secure Sketch for Point-Set Difference
A secure sketch is a set of published data that can help to recover the original biometric data after they are corrupted by permissible noises, and by itself does not reveal much information about the original. Several constructions have been proposed for different metrics, and in particular, set difference. We observe that in many promising applications, set difference alone is insufficient to model the noises. We propose to look into point-set difference, which measures noises that not only remove/introduce new feature points in the biometric objects, but may also perturb the points. In this paper, we first give an improvement for set difference construction that can be extended to multi-sets, where the sketch is small and there is an efficient decoding algorithm. We next give a sketch for point-set difference in both one and two-dimensional spaces. By using results in almost k-wise independence, the size of the sketch is reduced to near-optimal.