Verifiying that an item/device/individual is a member of a group is needed for many applications granting or refusing access to sensitive resources. Group membership verification is not about identifying first and then checking membership. Rather, being granted with access requires that the members of the group could be distinguished from non-members, but it does not require to distinguish members from one another.
Group membership verification protocols first enroll eligible signatures into a data structure stored at a server. Then, at verification time, the structure is queried by a client signature and the access is granted or not. For security, the data structure must be adequately protected so that a honest but curious server cannot reconstruct the signatures. For privacy, verification should proceed anonymously, not disclosing identities.
A client signature is a noisy version of the enrolled one, due to changes in lighting conditions e.g. The verification must absorb such variations and cope with the continuous nature of signatures. They must be such that it is unlikely that a noisy version for one user gets similar enough to the enrolled signature of any other user. Continuity, discriminability and statistical independence are inherent properties of signatures.
This paper proposes a group membership verification protocol preventing a curious but honest server from reconstructing the enrolled signatures and inferring the identity of querying (trusted) clients. It combines two building blocks:
One building block hashes continuous vectors into discreteembeddings. This lossy process limits the ability of the server to reconstruct signatures from the embeddings. Block #2: The other building block aggregates multiple vectors into a unique representative value which will be enrolled at the server. The server can therefore not infer any specific signature from this value. Sufficient information must be preserved through the aggregation process for the server to assert whether or not a querying signature is a member of the group.
These two blocks can be assembled according to two configurations: block #1 before block #2, the system acquires and then hashes the signatures before aggregating them. The opposite configuration is where acquired signatures are aggregated before hashing the result of this aggregation. At query time, the newly acquired signature is always hashed before being sent to the server. Weaknesses and strengths of these two configurations are explored in the paper.
2 Related Work
Group membership verification protocols relying on cryptography exist  but are more relevant to authentication, identification and secret binding. Other approaches apply homomorphic encryption to signatures, compare  and threshold them [3, 4] in the encrypted domain, and need active participation of clients. Approaches involving cryptography, however, are extremely costly, memory and CPU wise.
Group membership is linked to Bloom filters that are used to test whether an element is a member of a set. When considering security, a server using Bloom filters cannot infer any information on one specific entry . Note that Bloom filters can not deal with continuous high dimensional signatures and that queries must be encrypted to protect the privacy of users [6, 7]. Bloom filters, adapted to our setup, however, form a baseline in our experiments (see Sect. 5.3).
Embedding a single high dimensional signature is quite a standard technique. The closest to our work is the privacy-preserving identification mechanism based on sparsifying transform [8, 9]. It produces an information-preserving sparse ternary embedding, ensuring privacy of the data users and security of the signature.
Aggregating signals into similarity-preserving representations is very common in computer vision[10, 11, 12]. They do not consider security or privacy. In , Iscen et al. use the group testing paradigm to pack a random set of image signatures into a unique high-dimensional vector. It is therefore an excellent basis for the aggregation block: the similarities between the original non-aggregated signatures and a query signature is preserved through the aggregation.
3 Notations and definitions
Signatures are vectors in . If users/items belong to the group, then the protocol considers signatures, , …, . The signature to verify is a query vector . Group membership verification considers two hypotheses linked to the continuous nature of the signatures:
: The query is related to one of the vectors. For instance, it is a noisy version of vector , , with to be a noise vector.
: The query is not related to any vector in the group.
We first design a group aggregation technique which computes a single representation from all vectors . This is done at the enrollment phase. Variable denotes the size in bits of this representation.
At the verification phase, the query is hashed by a function of size in bits. This function might be probabilistic to ensure privacy. The group membership test decides which hypothesis is deemed true by comparing and . This is done by first computing a score function and thresholding its results: .
3.1 Verification Performances
The performances of this test are measured by the probabilities of false negative,, and false positive, . As varies from and , these measures are summarized by the AUC (Area Under Curve) performance score. Another figure of merit is for s.t. , a required false positive level.
3.2 Security and Privacy
A curious server can reconstruct a signature from its embedding (for instance the query): . The mean squared error is a way to assess its accuracy: The best reconstruction is known to be the conditional expectation: .
Reconstructing an enrolled signature from the group representation is even more challenging. Due to the aggregation block, the curious server can only reconstruct a single vector
from the aggregated representation, and this vector serves as an estimation of any signatures in the group:
4 Verification for a few group members
This section discusses the verification protocol when is small. We study the two different configurations for assembling block #1 and block #2.
Block #1: Embedding. An embedding maps a vector to a sequence of discrete symbols. This quantization shall preserve enough information to tell whether two embeddings are related, but not enough to reconstruct a signature. We use the sparsifying transform coding [8, 9]. It projects to the range space of a transform matrix . The output alphabet is imposed by quantizing the components of whose amplitude is lower than to 0, the others to +1 or -1 according to their sign. In expectation, symbols are non null.
Block #2: Aggregation. Aggregation processes a set of input vectors to produce a unique output vector. When block #1 is used before, that is, when considering , then . When block #2 is used before block #1, that is when considering , then .
4.1 Aggregation strategies
The nature of highly depends on the type of vector the aggregation function receives. When considering , then gets continuous signatures. In this case it is possible to design two aggregations schemes that are:
4.2 Four resulting schemes
The assemblage of the blocks and the aggregation strategies overall create four variants. We name them:
The score function comparing the hashed query with the group representation is always .
5 Reconstruction and Verification
This section makes the following assumptions: i) Enrolled signatures are modelled by
, ii) Square orthogonal matrixknown by the attacker.
5.1 Ability to reconstruct from the embedding
Now that preserves the norm, the on is the same as the mean square reconstruction error on
, which is also white Gaussian distributed. Thanks to the independance of the components of, the conditional expectation can be computed component-wise. We introduce the density function conditioned on the interval :
with intervals , , and . Function is the p.d.f. of and is the indicator function of interval .
Observing the -th symbol of equals reveals that . This component is reconstructed as . Note that because is symmetric around 0. For , the reconstruction value equals , where . By symmetry, , and admits the following close form:
This quantity starts at when . The embeddings are then full binary words (). All components are reconstructed by
but with a large variance. Asincreases, this variance decreases but less non-null components are reconstructed. achieves a minimum of for , where of the symbols of an embedding are non null. Then, increases up to for a large : the embeddings becomes sparser and sparser. When fully zero, each component is reconstructed by , and equals .
5.2 Ability to reconstruct the signatures
The curious server tries to reconstruct a unique vector from which represents the enrolled signatures. Note that is scale invariant: scaling the signatures by any positive factor does not change . Suppose that the curious server reconstructs . The best scaling minimizing (1) is: with . The curious server can not compute giving birth to a larger distortion:
This lower bound is further minimized by choosing .
The first term is the squared distance between and , whereas the second term corresponds to the error reconstruction for inverting the embedding. In the end:
This figure of merit increases with because , : Packing more signatures increases security.
5.3 Verification performances
We compare to a baseline defined as a Bloom filter optimally tuned for given and having length . An embedding is mandatory to first turn the real signatures into discrete objects. This means that, under , a false negative happens whenever .
Fig. (a)a shows the vs. (7) for the schemes of Sect. 4.1 for different sparsity . Two schemes performs better. For low privacy (small ), HoA-3 achieves the largest (with ) ; for high privacy, AoH-4 is recommended (with ). In these regimes, the performances are better than the Bloom filter.
6 Verification for multiple groups
When is large, aggregating all the signatures into a unique performs poorly. Rather, for large , we propose to partition the enrolled signature into groups, and to compute different representatives, one per partition.
Random assignment: The signatures are randomly assigned into groups of size .
Similar signatures are assigned to the same group. The paper uses the k-means algorithm to do so. Yet, the size of the groups is no longer constant.
6.1 Verification performances
Denote by the operating point of group number , . The overall system outputs a positive answer when at least one group test is positive. Denote by the performance of the global system. Under , the query is not related to any vector. Therefore,
Under , the query is related to only one vector belonging to one group. A false negative occurs, if this test produces a false negative and the other tests a true negative each:
The operating point of a group test is mainly due to the size of the group. The random assignment creates even groups (if divides ), so these share the operating point .
Fig. (a)a shows the experimental and the one predicted by (12) and (13) when ranges from to . Since clustering makes groups of different sizes, we show the performances versus , where is the size of -th group. The theoretical formulas are more accurate for random partitioning where the group are even. Estimations of were less precise with the clustering strategy, and this inaccuracy cumulates in (12) and (13).
Clustering improves the verification performances a lot especially for HoA-3. A similar phenomenon was observed in . Yet, Fig. (b)b shows that it does not endanger the system: is only slightly smaller than for random assignment, and indeed close to 1 for . This is obtained for for HoA-3 giving . The space is so big that the clusters are gigantic and not revealing much about where the signatures are. However, the anonymity is reduced because the server learns which group provided a positive test. This is measured in term of -anonymity by the size of the smallest group, i.e. . Fig. (a)a indeed shows the trade-off between -anonymity and the verification performances.
This paper proposed four schemes for verifying the group membership of continuous high dimensional vectors. The keystones are the aggregation and embedding functions. They prevent accurate reconstruction of the enrolled signatures, while recognizing noisy version. However, the anonymity is slightly revealed when managing many signatures aggregated into several representatives: the server is only able to link each signature to its group number. Yet, the full identity of the user is preserved.
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