Efficient reductions in cyclotomic rings - Application to Ring-LWE based FHE schemes

Abstract : With Fully Homomorphic Encryption (FHE), it is possible to process encrypted data without access to the private-key. This has a wide range of applications, most notably the offloading of sensitive data processing. Most research on FHE has focused on the improvement of its efficiency, namely by introducing of schemes based on Ring-Learning With Errors (RLWE), and techniques such as batching, which allows for the encryption of multiple messages in the same ciphertext. Much of the related research has focused on RLWE relying on power-of-two cy-clotomic polynomials. While it is possible to achieve efficient arithmetic with such polynomials, one cannot exploit batching. Herein, the efficiency of ring arithmetic underpinned by non-power-of-two cyclomotic polyno-mials is analyzed and improved. Two methods for polynomial reduction are proposed, one based on the Barrett reduction and the other on a Montgomery representation. Speed-ups of up to 2.66 are obtained for the reduction operation using an i7-5960X processor when compared with a straightforward implementation of the Barrett reduction. Moreover, the proposed methods are exploited to enhance homomorphic multiplication of Fan-Vercauteren (FV) and Brakerski-Gentry-Vaikuntantahan (BGV) encryption schemes, producing experimental speed-ups of up to 1.37.
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Jean-Claude Bajard, Julien Eynard, Anwar Hasan, Paulo Martins, Leonel Sousa, et al.. Efficient reductions in cyclotomic rings - Application to Ring-LWE based FHE schemes. Selected Areas of Cryptography 2017, Aug 2017, Ottawa, Canada. ⟨hal-01585516⟩

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