Note on the noise growth of the RNS variants of the BFV scheme

In a recent work, Al Badawi et al. have noticed a different behaviour of the noise growth in practice between the two RNS variants of BFV from Bajard et al. and Halevi et al. Their experiments, based on the PALISADE and SEAL libraries, have shown that the multiplicative depth reached, in practice, by the first one was considerably smaller than the second one while theoretically equivalent in the worst-case. Their interpretation of this phenomenon was that the approximations used by Bajard et al. made the expansion factor behave differently than what the Central Limit Theorem would predict. We have realized that this difference actually comes from the implementation of the SmMRq procedure of Bajard et al. in SEAL and PALISADE which is slightly different than what Bajard et al. had proposed. In this note we show that by fixing this small difference, the multiplicative depth of both variants is actually the same in practice

Towards the AlexNet Moment for Homomorphic Encryption: HCNN, theFirst Homomorphic CNN on Encrypted Data with GPUs

Deep Learning as a Service (DLaaS) stands as a promising solution for cloud-based inference applications. In this setting, the cloud has a pre-learned model whereas the user has samples on which she wants to run the model. The biggest concern with DLaaS is user privacy if the input samples are sensitive data. We provide here an efficient privacy-preserving system by employing high-end technologies such as Fully Homomorphic Encryption (FHE), Convolutional Neural Networks (CNNs) and Graphics Processing Units (GPUs). FHE, with its widely-known feature of computing on encrypted data, empowers a wide range of privacy-concerned applications. This comes at high cost as it requires enormous computing power. In this paper, we show how to accelerate the performance of running CNNs on encrypted data with GPUs. We evaluated two CNNs to classify homomorphically the MNIST and CIFAR-10 datasets. Our solution achieved a sufficient security level (> 80 bit) and reasonable classification accuracy (99%) and (77.55%) for MNIST and CIFAR-10, respectively. In terms of latency, we could classify an image in 5.16 seconds and 304.43 seconds for MNIST and CIFAR-10, respectively. Our system can also classify a batch of images (> 8,000) without extra overhead

Revisiting Homomorphic Encryption Schemes for Finite Fields

The Brakerski-Gentry-Vaikuntanathan (BGV) and Brakerski/ Fan-Vercauteren (BFV) schemes are the two main homomorphic encryption (HE) schemes to perform exact computations over finite fields and integers. Although the schemes work with the same plaintext space, there are significant differences in their noise management, algorithms for the core homomorphic multiplication operation, message encoding, and practical usability. The main goal of our work is to revisit both schemes, focusing on closing the gap between the schemes by improving their noise growth, computational complexity of the core algorithms, and usability. The other goal of our work is to provide both theoretical and experimental performance comparison of BGV and BFV. More precisely, we propose an improved variant of BFV where the encryption operation is modified to significantly reduce the noise growth, which makes the BFV noise growth somewhat better than for BGV (in contrast to prior results showing that BGV has smaller noise growth for larger plaintext moduli). We also modify the homomorphic multiplication procedure, which is the main bottleneck in BFV, to reduce its algorithmic complexity. Our work introduces several other novel optimizations, including lazy scaling in BFV homomorphic multiplication and an improved BFV decryption procedure in the Residue Number System (RNS) representation. We also develop a usable variant of BGV as a more efficient alternative to BFV for common practical scenarios. We implement our improved variants of BFV and BGV in PALISADE and evaluate their experimental performance for several benchmark computations. The experimental results suggest that our BGV implementation is faster for intermediate and large plaintext moduli, which are often used in practical scenarios with ciphertext packing, while our BFV implementation is faster for small plaintext moduli

Implementation and Performance Evaluation of RNS Variants of the BFV Homomorphic Encryption Scheme

Homomorphic encryption is an emerging form of encryption that provides the ability to compute on encrypted data without ever decrypting them. Potential applications include aggregating sensitive encrypted data on a cloud environment and computing on the data in the cloud without compromising data privacy. There have been several recent advances resulting in new homomorphic encryption schemes and optimized variants. We implement and evaluate the performance of two optimized variants, namely Bajard-Eynard-Hasan-Zucca (BEHZ) and Halevi-Polyakov-Shoup (HPS), of the most promising homomorphic encryption scheme in CPU and GPU. The most interesting (and also unexpected) result of our performance evaluation is that the HPS variant in practice scales significantly better (typically by 15%-30%) with increase in multiplicative depth of the computation circuit than BEHZ, implying that the HPS variant will always outperform BEHZ for most practical applications. For the multiplicative depth of 98, our fastest GPU implementation performs homomorphic multiplication in 51 ms for 128-bit security settings, which is faster by two orders of magnitude than prior results and already practical for cloud environments supporting GPU computations. Large multiplicative depths supported by our implementations are required for applications involving deep neural networks, logistic regression learning, and other important machine learning problems

OpenFHE: Open-Source Fully Homomorphic Encryption Library

Fully Homomorphic Encryption (FHE) is a powerful cryptographic primitive that enables performing computations over encrypted data without having access to the secret key. We introduce OpenFHE, a new open-source FHE software library that incorporates selected design ideas from prior FHE projects, such as PALISADE, HElib, and HEAAN, and includes several new design concepts and ideas. The main new design features can be summarized as follows: (1) we assume from the very beginning that all implemented FHE schemes will support bootstrapping and scheme switching; (2) OpenFHE supports multiple hardware acceleration backends using a standard Hardware Abstraction Layer (HAL); (3) OpenFHE includes both user-friendly modes, where all maintenance operations, such as modulus switching, key switching, and bootstrapping, are automatically invoked by the library, and compiler-friendly modes, where an external compiler makes these decisions. This paper focuses on high-level description of OpenFHE design, and the reader is pointed to external OpenFHE references for a more detailed/technical description of the software library

An Improved RNS Variant of the BFV Homomorphic Encryption Scheme

We present an optimized implementation of the Fan-Vercauteren variant of Brakerski\u27s scale-invariant homomorphic encryption scheme. Our algorithmic improvements focus on optimizing decryption and homomorphic multiplication in the Residue Number System (RNS), using the Chinese Remainder Theorem (CRT) to represent and manipulate the large coefficients in the ciphertext polynomials. In particular, we propose efficient procedures for scaling and CRT basis extension that do not require translating the numbers to standard (positional) representation. Compared to the previously proposed RNS design due to Bajard et al., our procedures are simpler and faster, and introduce a lower amount of noise. We implement our optimizations in the PALISADE library and evaluate the runtime performance for the range of multiplicative depths from 1 to 100. For example, homomorphic multiplication for a depth-20 setting can be executed in 62 ms on a modern server system, which is already practical for some outsourced-computing applications. Our algorithmic improvements can also be applied to other scale-invariant homomorphic encryption schemes, such as YASHE

Asymptotically Faster Multi-Key Homomorphic Encryption from Homomorphic Gadget Decomposition

Homomorphic Encryption (HE) is a cryptosytem that allows us to perform an arbitrary computation on encrypted data. The standard HE, however, has a disadvantage in that the authority is concentrated in the secret key owner since computations can only be performed on ciphertexts encrypted under the same secret key. To resolve this issue, research is underway on Multi-Key Homomorphic Encryption (MKHE), which is a variant of HE supporting computations on ciphertexts possibly encrypted under different keys. Despite its ability to provide privacy for multiple parties, existing MKHE schemes suffer from poor performance due to the cost of multiplication which grows at least quadratically with the number of keys involved. In this paper, we revisit the work of Chen et al. (ACM CCS 2019) on MKHE schemes from CKKS and BFV and significantly improve their performance. Specifically, we redesign the multi-key multiplication algorithm and achieve an asymptotically optimal complexity that grows linearly with the number of keys. Our construction relies on a new notion of gadget decomposition, which we call homomorphic gadget decomposition, where arithmetic operations can be performed over the decomposed vectors with guarantee of its functionality. Finally, we implement our MKHE schemes and demonstrate their benchmarks. For example, our multi-key CKKS multiplication takes only 0.5, 1.0, and 1.9 seconds compared to 1.6, 5.9, and 23.0 seconds of the previous work when 8, 16, and 32 keys are involved, respectively

Efficiency of Homomorphic Encryption Schemes

In 2009, Craig Gentry introduced the first fully homomorphic encryption scheme using bootstrapping. In the 13 years since, a large amount of research has gone into improving efficiency of homomorphic encryption schemes. This includes implementing leveled homomorphic encryption schemes for practical use, which are schemes that allow for some predetermined amount of additions and multiplications that can be performed on ciphertexts. These leveled schemes have been found to be very efficient in practice. In this thesis, we will discuss the efficiency of various homomorphic encryption schemes. In particular, we will see how to improve sizes of parameter choices in homomorphic encryption schemes with a variety of techniques to include modulus leveling and techniques of error bound control