Affiliation: Chinese Association for Cryptologic Research
Improved Bootstrapping for Approximate Homomorphic Encryption 📺
Since Cheon et al. introduced a homomorphic encryption scheme for approximate arithmetic (Asiacrypt ’17), it has been recognized as suitable for important real-life usecases of homomorphic encryption, including training of machine learning models over encrypted data. A follow up work by Cheon et al. (Eurocrypt ’18) described an approximate bootstrapping procedure for the scheme. In this work, we improve upon the previous bootstrapping result. We improve the amortized bootstrapping time per plaintext slot by two orders of magnitude, from $$\sim $$∼1 s to $$\sim $$∼0.01 s. To achieve this result, we adopt a smart level-collapsing technique for evaluating DFT-like linear transforms on a ciphertext. Also, we replace the Taylor approximation of the sine function with a more accurate and numerically stable Chebyshev approximation, and design a modified version of the Paterson-Stockmeyer algorithm for fast evaluation of Chebyshev polynomials over encrypted data.
Multi-Key Homomorphic Encryption from TFHE
In this paper, we propose a Multi-Key Homomorphic Encryption (MKHE) scheme by generalizing the low-latency homomorphic encryption by Chillotti et al. (ASIACRYPT 2016). Our scheme can evaluate a binary gate on ciphertexts encrypted under different keys followed by a bootstrapping.The biggest challenge to meeting the goal is to design a multiplication between a bootstrapping key of a single party and a multi-key RLWE ciphertext. We propose two different algorithms for this hybrid product. Our first method improves the ciphertext extension by Mukherjee and Wichs (EUROCRYPT 2016) to provide better performance. The other one is a whole new approach which has advantages in storage, complexity, and noise growth.Compared to previous work, our construction is more efficient in terms of both asymptotic and concrete complexity. The length of ciphertexts and the computational costs of a binary gate grow linearly and quadratically on the number of parties, respectively. We provide experimental results demonstrating the running time of a homomorphic NAND gate with bootstrapping. To the best of our knowledge, this is the first attempt in the literature to implement an MKHE scheme.
On the binary sequences with high $GF(2)$ linear complexities and low $GF(p)$ linear complexities
Klapper  showed that there are binary sequences of period $q^n-1$ ($q$ is a prime power $p^m$, $p$ is an odd prime) with the maximal possible linear complexity $q^n-1$ when considered as sequences over $GF(2)$, while the sequences have very low linear complexities when considered as sequences over $GF(p)$. This suggests that the binary sequences with high $GF(2)$ linear complexities and low $GF(p)$ linear complexities are note secure in cryptography. In this note we give some simple constructions of the binary sequences with high $GF(2)$ linear complexities and low $GF(p)$ linear complexities. We also prove some lower bounds on the $GF(p)$ linear complexities of binary sequences and a lower bound on the number of the binary sequences with high $GF(2)$ linear complexities and low $GF(p)$ linear complexities .