Multi-Adjustable Join Schemes With Adaptive Indistinguishably Security

A multi-adjustable join ($\text{M-Adjoin}$M-Adjoin) scheme [Khazaei-Rafiee, IEEE TDSC 2020], a generalization of $\text{Adjoin}$Adjoin scheme [Popa-Zeldovich, MIT CSAIL TR 2012], is a symmetric-key primitive that enables a user to securely outsource his database to an external server, and later to issue join queries for a list of columns. In [Rafiee-Khazaei, IEEE TDSC 2021], based on the previously defined security notions for $\text{Adjoin}$Adjoin [Mironov-Segev-Shahaf, TCC 2017], several security notions for $\text{M-Adjoin}$M-Adjoin were proposed and their relationships were investigated. Constructing an $\text{M-Adjoin}$M-Adjoin with indistinguishability security against adaptive adversary has remained a challenging problem so far. In this paper, we introduce two $\text{M-Adjoin}$M-Adjoin constructions to achieve this strong security notion in the random oracle model. We prove the security of our constructions under Decisional Diffie-Hellman assumption in $\mathbb {G}_{1}$G1 (DDH1) in the bilinear groups. Compared with previous constructions, despite having a higher security level, the computation and storage overheads do not increase.