  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.
