Search over Compute: Solving Multiplication-Intensive Computational Problems over FHE Data

Fully Homomorphic Encryption (FHE) is a crypto-system that enables computations on encrypted data without decrypting them, thus providing privacy. The practical applications of FHE have performance bottleneck due to the complexity of operations involved. FHE usually supports addition and multiplication as the two primitive operations, which further enable arbitrary computations on encrypted data. Of these, multiplication is computationally expensive operation in homomorphic domain when compared to addition. Hence, computations involving high number of multiplications have performance issues. In this paper, we improve the performance of a narrow class of multiplication intensive computational problems in encrypted domain that have finite result space. We solve such problems using search on publicly known unencrypted database of result space instead of computing the problem using traditional methods over encrypted data. This class of multiplication-intensive computations can be solved efficiently by searching due to fewer number of multiplications involved when compared to traditional computation, providing high performance improvement. Furthermore, we improve the performance of search algorithm by exploiting boolean arithmetic when search is performed over a public dataset. We illustrate this two-level optimization using factorial computation on encrypted data and demonstrate that a simple optimization of solving factorial as a search problem shows enormous performance improvement when compared to direct computation in encrypted domain. Factorial is an important primitive in several disciplines such as combinatorics, probability theory etc. which are building blocks of cloud analytics. We demonstrate through experiments that computation of factorial efficiently in turn improves the performance of such applications considerably. This optimization can be used to improve performance of other multiplication-intensive computational problems like exponentiation, binomial computations, and other domain specific computations.