Approximate Homomorphic Encryption with Reduced Approximation Error

Kim, Andrey; Papadimitriou, Antonis; Polyakov, Yuriy

doi:10.1007/978-3-030-95312-6_6

Cited by 27 publications

(21 citation statements)

References 16 publications

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“…We apply several methods to reduce the rescaling error and relinearization error and ensure the precision of the resultant message, such as the scaling factor management in [30], lazy rescaling, and lazy relinearization [31], [32]. The lazy rescaling and relinearization can also be applied to reduce the computation time as it requires much computation due to the number-theoretic transformation (NTT) and gadget decomposition.…”

Section: ) Optimization For Precision Of Homomorphic Operationsmentioning

confidence: 99%

Privacy-Preserving Machine Learning With Fully Homomorphic Encryption for Deep Neural Network

et al. 2022

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Fully homomorphic encryption (FHE) is one of the prospective tools for privacy-preserving machine learning (PPML), and several PPML models have been proposed based on various FHE schemes and approaches. Although the FHE schemes are known as suitable tools to implement PPML models, previous PPML models on FHE such as CryptoNet, SEALion, and CryptoDL are limited to only simple and non-standard types of machine learning models. These non-standard machine learning models are not proven efficient and accurate with more practical and advanced datasets. Previous PPML schemes replace non-arithmetic activation functions with simple arithmetic functions instead of adopting approximation methods and do not use bootstrapping, which enables continuous homomorphic evaluations. Thus, they could not use standard activation functions and could not employ a large number of layers. In this work, we firstly implement the standard ResNet-20 model with the RNS-CKKS FHE with bootstrapping and verify the implemented model with the CIFAR-10 dataset and the plaintext model parameters. Instead of replacing the non-arithmetic functions with the simple arithmetic function, we use state-of-the-art approximation methods to evaluate these non-arithmetic functions, such as the 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 the 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 about 3 hours for inference on a dual Intel Xeon Platinum 8280 CPU (112 cores) with 172 GB memory. We think that it opens the possibility of applying the FHE to the advanced deep PPML model.

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Section: ) Optimization For Precision Of Homomorphic Operationsmentioning

confidence: 99%

Privacy-Preserving Machine Learning With Fully Homomorphic Encryption for Deep Neural Network

et al. 2022

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“…Moreover, a different kind of complex packing was introduced by Kim and Song in [131]. In [132], Kim, Papadimitriou and Polyakov improved the usability of CKKS and its RNS variant proposing a new technique that minimizes the error during computation. Specifically, the idea is to rescale the ciphertext before multiplication and not after, thus obtaining a smaller error before performing the multiplication.…”

Section: F Fourth Generation: Fhe Based On Lwe and Rlwementioning

confidence: 99%

“…The bootstrapping version of [128] includes an homomorphic modular reduction, which is approximated by a trigonometric function to improve efficiency. Parallel works improved this approximation, such as Lee, Lee, Lee, Kim and No [135], who improved the bootstrapping of the RNS-CKKS leveraging the technique proposed in [132]. Also, Jutla and Manohar [136] proposed a sine series to approximate the modular reduction, and achieved a significantly higher precision than the previous works.…”

Section: F Fourth Generation: Fhe Based On Lwe and Rlwementioning

confidence: 99%

Survey on Fully Homomorphic Encryption, Theory, and Applications

Marcolla

Sucasas

Manzano³

et al. 2022

Proc. IEEE

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Data privacy concerns are increasing significantly in the context of Internet of Things, cloud services, edge computing, artificial intelligence applications, and other applications enabled by next generation networks. Homomorphic Encryption addresses privacy challenges by enabling multiple operations to be performed on encrypted messages without decryption. This paper comprehensively addresses homomorphic encryption from both theoretical and practical perspectives. The paper delves into the mathematical foundations required to understand fully homomorphic encryption (FHE). It consequently covers design fundamentals and security properties of FHE, and describes the main FHE schemes based on various mathematical problems. On a more practical level, the paper presents a view on privacypreserving Machine Learning using homomorphic encryption, then surveys FHE at length from an engineering angle, covering the potential application of FHE in fog computing, and cloud computing services. It also provides a comprehensive analysis of existing state-of-the-art FHE libraries and tools, implemented in software and hardware, and the performance thereof.

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“…A technique of eliminating the large rescaling error in the RNS-CKKS scheme was proposed in [27], where different scaling factors in different levels were used instead of using the same scaling factor for each level. If the maximum level is L, and the ciphertext modulus for level i is q i , the scaling factor for each level is set as follows:…”

Section: Scaling Factor Managementmentioning

confidence: 99%

Optimization of Homomorphic Comparison Algorithm on RNS-CKKS Scheme

Lee

Kim

et al. 2022

IEEE Access

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Since the sign function can be used to implement the comparison operation, max function, and rectified linear unit (ReLU) function, several studies have been conducted to efficiently evaluate the sign function in the Cheon-Kim-Kim-Song (CKKS) scheme, one of the most promising fully homomorphic encryption schemes. Recently, Lee et al. (IEEE Trans. Depend. Sec. Comp.) proposed a practically optimal approximation method of sign function on the CKKS scheme using a composition of minimax approximate polynomials. In addition, Lee et al. proposed a polynomial-time algorithm that finds degrees of component polynomials minimizing the number of non-scalar multiplications for homomorphic comparison/max/ReLU functions. However, homomorphic comparison/max/ReLU functions using Lee et al.'s approximation method have not been successfully implemented on the residue number system variant CKKS (RNS-CKKS) scheme. In addition, the degrees of component polynomials found by Lee et al.'s algorithm are not optimized for the RNS-CKKS scheme because the algorithm does not consider that the running time of non-scalar multiplication depends much on the ciphertext level in the RNS-CKKS scheme. In this paper, we propose a fast algorithm for inverse minimax approximation error, a subroutine required to find the optimal set of degrees of component polynomials. This proposed algorithm makes it possible to find the optimal set of degrees of component polynomials with higher degrees than the previous study. In addition, we propose a method to find the degrees of component polynomials optimized for the RNS-CKKS scheme using the proposed algorithm for inverse minimax approximation error. We successfully implement the homomorphic comparison, max function, and ReLU function algorithms on the RNS-CKKS scheme with a low comparison failure rate (< 2 −15 ) and provide the various parameter sets according to the precision parameter α. We reduce the depth consumption of the homomorphic comparison, max function, and ReLU function algorithms by one depth for several α. In addition, the numerical analysis demonstrates that the homomorphic comparison, max function, and ReLU function algorithms using the degrees of component polynomials found by the proposed algorithm reduce running time by 6%, 7%, and 6% on average compared with those using the degrees of component polynomials found by Lee et al.'s algorithm, respectively.INDEX TERMS Cheon-Kim-Kim-Song (CKKS) scheme, fully homomorphic encryption (FHE), homomorphic comparison operation, minimax approximate polynomial, Remez algorithm, residue number system variant CKKS (RNS-CKKS) scheme.

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Approximate Homomorphic Encryption with Reduced Approximation Error

Cited by 27 publications

References 16 publications

Privacy-Preserving Machine Learning With Fully Homomorphic Encryption for Deep Neural Network

Privacy-Preserving Machine Learning With Fully Homomorphic Encryption for Deep Neural Network

Survey on Fully Homomorphic Encryption, Theory, and Applications

Optimization of Homomorphic Comparison Algorithm on RNS-CKKS Scheme

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