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Table 1 Complexity analysis of FHE-BLOOM and PHE-BLOOM: Setup overheads are similar in both approaches and grow linearly in the number of patients n and Bloom filter size l which is proportional to the number of SNPs m, i.e., l=−m log(p)/ log(2)2

From: BLOOM: BLoom filter based oblivious outsourced matchings

Approach DB setup (Client) Query (Cloud) Query (Client)
  Time Comm Time Time Comm.
Fhe-Bloom \(\mathcal {O}(n \cdot l / s_{F})~\text {Enc}_{F}\) \(\mathcal {O}(n \cdot l / s_{F})~\mathrm {C}_{F}\) \(\mathcal {O}(n \cdot l / s_{F})~\text {Mul}_{F} + \mathcal {O}(n \cdot l / s_{F})~\text {Add}_{F}\) \(\mathcal {O}(l / s_{F}) ~\text {Enc}_{F} +\mathcal {O}(n) ~ \text {Dec}_{F}\) \(\mathcal {O}(l / s_{F}+n) ~\mathrm {C}_{F}\)
Phe-Bloom \(\mathcal {O}(n \cdot l / s_{P})~\text {Enc}_{P}\) \(\mathcal {O}(n \cdot l / s_{P})~\mathrm {C}_{P}\) \(\mathcal {O}(n / s_{P}) ~\text {Add}_{P}\) \(\mathcal {O}(n / s_{P}) ~\text {Dec}_{P}\) \(\mathcal {O}(n / s_{P}) ~ \mathrm {C}_{P}\)