Fully homomorphic encryption (FHE) is a prospective tool for privacy-preserving machine learning (PPML). Several PPML models have been proposed based on various FHE schemes and approaches. Although FHE schemes are suitable as tools for implementing PPML models, previous PPML models based on FHE, such as CryptoNet, SEALion, and CryptoDL, are limited to simple and nonstandard types of machine learning models; they have not proven to be efficient and accurate with more practical and advanced datasets. Previous PPML schemes replaced non-arithmetic activation functions with simple arithmetic functions instead of adopting approximation methods and did not use bootstrapping, which enables continuous homomorphic evaluations. Thus, they could neither use standard activation functions nor employ large numbers of layers. In this work, we first implement the standard ResNet-20 model with the RNS-CKKS FHE with bootstrapping and verify the implemented model with the CIFAR-10 dataset and plaintext model parameters. Instead of replacing the non-arithmetic functions with simple arithmetic functions, we use state-of-the-art approximation methods to evaluate these non-arithmetic functions, such as ReLU and Softmax, with sufficient precision. Further, for the first time, we use the bootstrapping technique of the RNS-CKKS scheme in the proposed model, which enables us to evaluate an arbitrary deep learning model on encrypted data. We numerically verify that the proposed model with the CIFAR-10 dataset shows 98.43% identical results to the original ResNet-20 model with non-encrypted data. The classification accuracy of the proposed model is 92.43%±2.65%, which is quite close to that of the original ResNet-20 CNN model (91.89%). It takes approximately 3 h for inference on a dual Intel Xeon Platinum 8280 CPU (112 cores) with 172 GB of memory. We believe that this opens the possibility of applying FHE to an advanced deep PPML model.

Homomorphic Encryption (HE) makes it possible to compute on encrypted data without decryption. In lattice-based HE, a ciphertext contains noise, which accumulates along with homomorphic computations. Bootstrapping refreshes the noise and it is possible to perform arbitrary-depth computations on HE with bootstrapping, which we call Fully Homomorphic Encryption (FHE). In this article, we propose a new general bootstrapping technique for RLWE-based schemes and its practical instantiation for FHE. It can be applied to all three RLWE-based leveled FHE schemes: Brakerski-Gentry-Vaikuntanathan (BGV), Brakerski/Fan-Vercauteren (BFV), and Cheon-Kim-Kim-Song (CKKS) with minor deviations in the algorithms. Our new construction of bootstrapping extracts a noiseless ciphertext for a part of the input, scales it, and finally removes it. In contrast with previous bootstrapping algorithms, the proposed method consumes only 1–2 levels and uses smaller parameters. For BGV and BFV, our new bootstrapping does not have any restrictions on a plaintext modulus unlike typical cases of the previous methods. The error introduced by our approach for CKKS is comparable to a rescaling error, allowing us to preserve a large amount of precision after bootstrapping.

Since Cheon et al. introduced an approximate homomorphic encryption scheme for complex numbers called Cheon-Kim-Kim-Song (CKKS) scheme, it has been widely used and applied in real-life situations, such as privacy-preserving machine learning. The polynomial approximation of a modulus reduction is the most difficult part of the bootstrapping for the CKKS scheme. In this paper, we cast the problem of finding an approximate polynomial for a modulus reduction into an L2-norm minimization problem. As a result, we find an approximate polynomial for the modulus reduction without using the sine function, which is the upper bound for the approximation of the modulus reduction. With the proposed method, we can reduce the degree of the polynomial required for an approximate modulus reduction, while also reducing the error compared with the most recent result reported by Han et al. (CT-RSA’ 20). Consequently, we can achieve a low-error approximation, such that the maximum error is less than 2−40 for the size of the message m/q ≈ 2−10. By using the proposed method, the constraint of q = O(m3/2) is relaxed as O(m), and thus the level loss in bootstrapping can be reduced. The solution to the cast problem is determined in an efficient manner without iteration.

We present a novel code-based signature scheme called modified pqsigRM. This scheme is based on a modified Reed–Muller (RM) code, which reduces the signing complexity and key size compared with existing code-based signature schemes. In fact, it strengthens pqsigRM submitted to NIST for post-quantum cryptography standardization. The proposed scheme has the advantage of the pqsigRM decoder and uses public codes that are more difficult to distinguish from random codes. We use (U,U + V)-codes with the high-dimensional hull to overcome the disadvantages of code-based schemes. The proposed decoder samples from coset elements with small Hamming weight for any given syndrome and efficiently finds such an element. Using a modified RM code, the proposed signature scheme resists various known attacks on RM-code-based cryptography. For 128 bits of classical security, the signature size is 4096 bits, and the public key size is less than 1 MB.