Towards More Secure Constructions of Adjustable Join Schemes

An adjustable join ( <inline-formula><tex-math notation="LaTeX">$\text{Adjoin}$</tex-math></inline-formula> ) scheme [4] is a symmetric-key primitive that enables a user to securely outsource his database to a server, and later to issue join queries for a pair of columns. When queries are extended to a list of columns, the <inline-formula><tex-math notation="LaTeX">$\mathtt{3Partition}$</tex-math></inline-formula> security of Adjoin schemes [8] does not capture the expected security. To address this deficiency, we introduce the syntax and security notion of multi-adjustable join ( <inline-formula><tex-math notation="LaTeX">$\text{M-Adjoin}$</tex-math></inline-formula> ) schemes. We propose a new security notion for this purpose, which we refer to as <inline-formula><tex-math notation="LaTeX">$\mathtt{M3Partition}$</tex-math></inline-formula> . The <inline-formula><tex-math notation="LaTeX">$\mathtt{3Partition}$</tex-math></inline-formula> security of <inline-formula><tex-math notation="LaTeX">$\text{Adjoin}$</tex-math></inline-formula> extends to the <inline-formula><tex-math notation="LaTeX">$\mathtt{M3Partition}$</tex-math></inline-formula> security of <inline-formula><tex-math notation="LaTeX">$\text{M-Adjoin}$</tex-math></inline-formula> in a straightforward way. The gap between <inline-formula><tex-math notation="LaTeX">$\mathtt{3Partition}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathtt{M3Partition}$</tex-math></inline-formula> is filled with a sequence <inline-formula><tex-math notation="LaTeX">$\lbrace \mathtt{M3P}_{k}\rbrace _{k\in \mathbb {N}}$</tex-math></inline-formula> of security definitions where <inline-formula><tex-math notation="LaTeX">$\mathtt{M3P}_{1}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathtt{M3P}_{\infty }$</tex-math></inline-formula> , respectively, correspond to <inline-formula><tex-math notation="LaTeX">$\mathtt{3Partition}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathtt{M3Partition}$</tex-math></inline-formula> . We propose constructions for achieving both <inline-formula><tex-math notation="LaTeX">$\mathtt{M3Partition}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathtt{M3P}_{k}$</tex-math></inline-formula> security levels. Our <inline-formula><tex-math notation="LaTeX">$\mathtt{M3Partition}$</tex-math></inline-formula> -secure scheme joins <inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula> columns, each containing <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> elements, in time <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(n^{m-1})$</tex-math></inline-formula> . Our <inline-formula><tex-math notation="LaTeX">$\mathtt{M3P}_{k}$</tex-math></inline-formula> -secure scheme uses ideas from secret sharing in its construction and does the job in time <inline-formula><tex-math notation="LaTeX">$\mathcal {O}\big ((m-1)n^k/k\big)$</tex-math></inline-formula> . It remains open if this barrier is inherent to the security definitions. Our schemes are substantially more efficient than the previous ones.