Group membership verification checks if a biometric trait corresponds to one member of a group without revealing the identity of that member. Recent contributions provide privacy for group membership protocols through the joint use of two mechanisms: quantizing templates into discrete embeddings and aggregating several templates into one group representation. However, this scheme has one drawback: the data structure representing the group has a limited size and cannot recognize noisy queries when many templates are aggregated. Moreover, the sparsity of the embeddings seemingly plays a crucial role on the performance verification. This paper proposes a mathematical model for group membership verification allowing to reveal the impact of sparsity on both security, compactness, and verification performances. This model bridges the gap towards a Bloom filter robust to noisy queries. It shows that a dense solution is more competitive unless the queries are almost noiseless.
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