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In biometric identification, a fingerprint is typically represented as a set of minutiae which are 2D points. A method [4] to protect the fingerprint template hides the minutiae by adding random points (known as chaff ) into the original point set. The chaff points are added one-by-one, constrained by the requirement that no two points are close to each other, until it is impossible to add more points or sufficient number of points have been added. Therefore, if the original template consists of s points, and the total number of chaff points and the original points is m, then a bruteforce attacker is expected to examine half of m chooses s possibilities to find the original. The chaff generated seem to be "random", especially if the minutiae are also randomly generated in the same manner. Indeed, the number of searches required by the brute-force attacker has been used to measure the security of the method. In this paper, we give an observation which leads to a way to distinguish the minutiae from the chaff. Extensive simulations show that our attacker can find the original better than brute-force search. For e.g. when s = 1 and the number of chaff points is expected to be about 313, our attacker on average takes about 100 searches. Our results highlight the need to adopt a more rigorous notion of security for template protection. We also give an empirical lower bound of the entropy loss due to the sketch.

Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed (cf. [4, 7, 19]). In this paper, generalising the usual homology of simplicial complexes, we define the embedded homology of hypergraphs as well as the persistent embedded homology of sequences of hypergraphs. As a generalisation of the Mayer-Vietoris sequence for the homology of simplicial complexes, we give a Mayer-Vietoris sequence for the embedded homology of hypergraphs. Moreover, as applications of the embedded homology, we study acyclic hypergraphs and construct some indices for the data analysis of hyper-networks.

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In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. For applications, we show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. We also prove a theorem that allows us to calculate the mod p 2 weighted persistent homology given some information on the mod p weighted persistent homology.2010 Mathematics Subject Classification. Primary 55N35, 55T99; Secondary 55U20, 55U10.