Fully homomorphic encryption over the integers for non-binary plaintexts without the sparse subset sum problem
- Authors
- Aung, K.M.M.; Lee, H.T.; Tan, B.H.M.; Wang, H.
- Issue Date
- Jun-2019
- Publisher
- Elsevier B.V.
- Keywords
- Bootstrapping; Fully homomorphic encryption; Non-binary plaintexts
- Citation
- Theoretical Computer Science, v.771, pp 49 - 70
- Pages
- 22
- Journal Title
- Theoretical Computer Science
- Volume
- 771
- Start Page
- 49
- End Page
- 70
- URI
- https://scholarworks.bwise.kr/cau/handle/2019.sw.cau/63738
- DOI
- 10.1016/j.tcs.2018.11.014
- ISSN
- 0304-3975
1879-2294
- Abstract
- In this work, we solve the open problem of designing a fully homomorphic encryption scheme over the integers for non-binary plaintexts in Z Q for prime Q (Q-FHE-OI) without the hardness of the sparse subset sum problem (SSSP). Furthermore, we show that our Q-FHE-OI scheme is a useful optimization for evaluating arithmetic circuits on encrypted data for some primes. To that end, we provide a natural extension of the somewhat homomorphic encryption (SHE) scheme over the integers proposed by Cheon and Stehlé (Eurocrypt 2015) to support non-binary plaintexts. Then, a novel bootstrapping algorithm is proposed for this extended SHE scheme by introducing generalizations of several functions in binary arithmetic. As a result, we obtain a Q-FHE-OI scheme for any constant-sized prime Q≥3 without the hardness of the SSSP, whose bootstrapping algorithm is asymptotically as efficient as previous best results. Beyond that, we compare the efficiency of our scheme against a Q-FHE-OI scheme obtained by emulating mod-Q gates with boolean circuits as proposed by Kim and Tibouchi (CANS 2016). Our analysis indicates our proposed scheme performs better for prime Q up to 11287, which improves on the result of Kim and Tibouchi, who showed there is at most one prime, Q=3 where the Q-FHE-OI scheme by Nuida and Kurosawa (Eurocrypt 2015) is a better approach. This overturns our previous understanding that Q-FHE-OI schemes do not provide significant benefits.
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