# Semantically Secure Order-Revealing Encryption: Multi-input Functional Encryption Without Obfuscation

**Authors:**

*D. Boneh, K. Lewi, M. Raykova, A. Sahai, M. Zhandry, and J. Zimmerman*

** Abstract: **

Deciding "greater-than" relations among data items just given their
encryptions is at the heart of search algorithms on encrypted data,
most notably, non-interactive binary search on encrypted data.
Order-preserving encryption provides one solution, but provably provides
only limited security guarantees. Two-input functional encryption is
another approach, but requires the full power of obfuscation machinery
and is currently not implementable.
We construct the first implementable encryption system supporting
greater-than comparisons on encrypted data that provides the
"best-possible" semantic security. In our scheme there is a public algorithm
that given two ciphertexts as input, reveals the order of the
corresponding plaintexts and nothing else. Our constructions are
inspired by obfuscation techniques, but do not use obfuscation. For
example, to compare two 16-bit encrypted values (e.g., salaries or age)
we only need a 9-way multilinear map. More generally, comparing
k-bit values requires only a (k/2+1)-way multilinear map. The
required degree of multilinearity can be further reduced, but at the
cost of increasing ciphertext size.
Beyond comparisons, our results give an implementable secret-key
multi-input functional encryption scheme for functionalities that can be
expressed as (generalized) branching programs of polynomial length and width.
Comparisons are a special case of this class, where for
k-bit inputs the branching program is of length k+1 and width 4.

** Reference:**

In proceedings of Eurocrypt 2015, LNCS 9057, pp. 563-594

**Full paper:**
pdf

**Related:** see our order-revealing encryption page.