## Cryptology ePrint Archive: Report 2019/332

Efficient Private Comparison Queries over Encrypted Databases using Fully Homomorphic Encryption with Finite Fields

Benjamin Hong Meng Tan and Hyung Tae Lee and Huaxiong Wang and Shu Qin Ren and Khin Mi Mi Aung

Abstract: To achieve security and privacy for data stored on the cloud, we need the ability to secure data in compute. Equality comparisons, $x=y, x\neq y$'', have been widely studied with many proposals but there is much room for improvement for order comparisons, $x < y,~x \leq y, x > y$ and $x \geq y$''. Most protocols for order comparisons have some limitation, either leaking some information about the data or requiring several rounds of communication between client and server. In addition, little work has been done on retrieving with compound conditions, mixing several equality and order comparisons. Fully homomorphic encryption (FHE) promises the ability to compute arbitrary functions on encrypted data without sacrificing privacy and without communication, but its potential has yet to be fulfilled. Particularly, private comparisons for database queries using FHE are expensive to compute.

In this work, we design efficient private database query (PDQ) protocols which support order comparisons and compound conditions. To this end, we first present a private comparison algorithm on encrypted integers using FHE, which scales efficiently for the length of input integers, by applying techniques from finite field theory. Then, we consider two scenarios for PDQ protocols, the first for retrieving data based on one order comparison and the second based on a conjunction of one order and four equality conditions. The proposed algorithm and protocols are implemented and tested to determine their performance in practice. The proposed comparison algorithm takes about 20.155 seconds to compare 697 pairs of 64-bit integers using Brakerski-Gentry-Vaikuntanathan's leveled FHE scheme with single instruction multiple data (SIMD) techniques at more than 110 bits of security. This yields an amortized rate of just 29 milliseconds per comparison. On top of that, we show that our techniques achieve an efficient PDQ protocol for one order and four equality comparisons, achieving an amortized time and communication cost of 36 milliseconds and 154 bytes per database element.

Category / Keywords: applications / private queries, encrypted database, homomorphic encryption