International Association
for Cryptologic Research

Recent attacks on NTRU lattices given by Ducas and van Woerden (ASIACRYPT 2021) showed that for moduli $q$ larger than the so-called fatigue point $n^{2.484+o(1)}$, the security of NTRU is noticeably less than that of (ring)-LWE. Unlike NTRU-based PKE with $q$ typically lying in the secure regime of NTRU lattices (i.e., $q<n^{2.484+o(1)}$), the security of existing NTRU-based multi-key FHEs (MK-FHEs) requiring $q=O(n^k)$ for $k$ keys could be significantly affected by those attacks. In this paper, we first propose a (matrix) NTRU-based MK-FHE for super-constant number $k$ of keys without using overstretched NTRU parameters. Our scheme is essentially a combination of two components following the two-layer framework of TFHE/FHEW: - a simple first-layer matrix NTRU-based encryption which naturally supports multi-key NAND operations with moduli $q=O(k\cdot n^{1.5})$ only linear in the number $k$ of keys; - and a crucial second-layer NTRU-based encryption which supports efficient hybrid product between a single-key ciphertext and a multi-key ciphertext for gate bootstrapping. Then, by replacing the first-layer with a more efficient LWE-based multi-key encryption, we obtain an improved MK-FHE scheme with better performance. We also employ a light key-switching technique to reduce the key-switching key size from previous $O(n^2)$ bits to $O(n)$ bits. A proof-of-concept implementation shows that our two MK-FHE schemes outperform the state-of-the-art TFHE-like MK-FHE schemes in both computation efficiency and bootstrapping key size. Concretely, for $k=8$ at the same 100-bit security level, our improved MK-FHE scheme can bootstrap a ciphertext in {0.54s} on a laptop and only has a bootstrapping key of size {13.89}MB, which are respectively 2.2 times faster and 7.4 times smaller than the MK-FHE scheme (which relies on a second-layer encryption from the ring-LWE assumption) due to Chen, Chillotti and Song (ASIACRYPT 2019).

Blind rotation is one of the key techniques to construct fully homomorphic encryptions with the best known bootstrapping algorithms running in less than one second. Currently, the two main approaches, namely, AP and GINX, for realizing blind rotation are first introduced by Alperin-Sheriff and Peikert (CRYPTO 2014) and Gama, Izabachene, Nguyen and Xie (EUROCRYPT 2016), respectively. In this paper, we propose a new blind rotation algorithm based on a GSW-like encryption from the NTRU assumption. Our algorithm has performance asymptotically independent from the key distributions, and outperforms AP and GINX in both the evaluation key size and the computational efficiency (especially for large key distributions). By using our blind rotation algorithm as a building block, we present new bootstrapping algorithms for both LWE and RLWE ciphertexts. We implement our bootstrapping algorithm for LWE ciphertexts, and compare the actual performance with two bootstrapping algorithms, namely, FHEW/AP by Ducas and Micciancio (EUROCRYPT 2015) and TFHE/GINX by Chillotti, Gama, Georgieva and Izabach\`ene (Journal of Cryptology 2020), that were implemented in the OpenFHE library. For parameters with ternary key distribution at 128-bit security, our bootstrapping only needs to store evaluation key of size 18.65MB for blind rotation, which is about 89.8 times smaller than FHEW/AP and 2.9 times smaller than TFHE/GINX. Moreover, our bootstrapping can be done in 112ms on a laptop, which is about 3.2 times faster than FHEW/AP and 2.1 times faster than TFHE/GINX. More improvements are available for large key distributions such as Gaussian distributions.