Dear Manoj Pokharel,

The following data may be helpful:

Homomorphic encryption is the Swiss army of cryptography. It allows to perform computations on encrypted data. this conjecture of processing on encrypted data was stated by [RAD78]. Until Gentry breakthrough in 2009, only partial solutions were offered. They deal with encrypted data with bounded functions in operations. Gentry raised the bound of number of operations by introducing a new method called the bootstrapping. This method allows to reduce the noise in the ciphertext and to perform more computing on it. Since Gentry's breakthrough, several improvements and several alternatives to the bootstrapping technique have been proposed to improve execution time and reduce resource consumption. This article studies the growth of noise and the noise management strategy in homomorphic encryption. It also presents the trend of hoping strategy from 2009 to 2016. Through the DGHV, it shows the management of noise on a one-bit encrypted message.

https://www.researchgate.net/publication/356758589_Noise_Management_in_Homomorphic_Encryption_Scheme_Analyzing_of_Algorithm

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A Regev-Type Fully Homomorphic Encryption Scheme Using Modulus Switching

A critical challenge in a fully homomorphic encryption (FHE) scheme is to manage noise. Modulus switching technique is currently the most efficient noise management technique. When using the modulus switching technique to design and implement a FHE scheme, how to choose concrete parameters is an important step, but to our best knowledge, this step has drawn very little attention to the existing FHE researches in the literature. The contributions of this paper are twofold. On one hand, we propose a function of the lower bound of dimension value in the switching techniques depending on the LWE specific security levels. On the other hand, as a case study, we modify the Brakerski FHE scheme (in Crypto 2012) by using the modulus switching technique. We recommend concrete parameter values of our proposed scheme and provide security analysis. Our result shows that the modified FHE scheme is more efficient than the original Brakerski scheme in the same security level.

https://www.hindawi.com/journals/tswj/2014/983862/

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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.

https://eprint.iacr.org/2021/204.pdf