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The Scientific World JOURNAL

A Regev-Type Fully Homomorphic Encryption Scheme Using Modulus Switching

Zhigang Chen, Jian Wang, Liqun Chen, Xinxia Song

Published Jan 1, 2014• 7 citations
Publisher

Abstract

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.

DOI
10.1155/2014/983862

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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.
There has been a great deal of work on improving the efficiency of fully homomorphic encryption (FHE) scheme. Our approach, in this regard, is to use the idea of packed ciphertexts to construct a multi-bit FHE with a short public key on the basis of the learning with errors (LWE) problem. More specifically, our FHE scheme builds on a basic encryption scheme that chooses LWE samples from the Gaussian distribution and adds Gaussian error to it. This results in decreasing the number of LWE samples from 2nlogq to n + 1. We prove that our FHE scheme is pragmatically feasible and its security relies on the hardness of the LWE problem. In addition, we form a new process of key switching for multi-bit FHE based on the ideas adopted by Brakerski et al. for optimizing the process of key switching. Finally, we analyze and compare the concrete parameters between our FHE scheme and BGH13 scheme. The result shows that compared with the BGH13 scheme, our scheme has a smaller public key by a factor about logq.
Abstract:To optimize noise management in fully homomorphic encryption scheme,a fully homomorphic encryption scheme with better parameters size was proposed.The security parameters of the scheme were estimated.Given security level,the maximal value of modulus q was got.Then the concert security parameters of the scheme were obtained by the condition of correct decryption among noise growth,circuit depth L and modulus q.The security parameters of Bra12 scheme also was obtained by the same way.The data showed that this scheme has better parameters size.The ciphertext from homomorphic multiplication is smaller by a factor about lb2 q than in Bra12 scheme and the corresponding tensored key is smaller by a factor about lb3 q than in Bra12 scheme.The matrix of key switching is smaller by a factor about lb2 q than in Bra12 scheme.The optimization for noise management is general,which can be used in the fully homomorphic encryption schemes based on LWE in which the noise growth depends on the secret key.
Fully homomorphic encryption (FHE) has important applications in cloud computing. However, almost all fully homomorphic encryption schemes share two common flaws that they all use large-scale secret keys and some operations are inefficient. In this paper, the 'special b' variant of the learning with errors problem (bLWE) is presented, and helps us construct the first circularly secure key switching process which can replace the key switching process and similar re-linearisation process used by the existing FHE schemes. Then, we present an efficient FHE. Compared with Brakerski's scheme, our scheme reduces L secret keys to one and is more efficient. Finally, we prove the chosen-plaintext attack (CPA) security of the fully homomorphic scheme and the circular security of key switching process in standard model under the learning with errors problem (LWE) assumption.
Fully homomorphic encryption (FHE) has important applications in cloud computing. However, almost all fully homomorphic encryption schemes share two common flaws that they all use large-scale secret keys and some operations are inefficient. In this paper, the 'special b' variant of the learning with errors problem (bLWE) is presented, and helps us construct the first circularly secure key switching process which can replace the key switching process and similar re-linearisation process used by the existing FHE schemes. Then, we present an efficient FHE. Compared with Brakerski's scheme, our scheme reduces L secret keys to one and is more efficient. Finally, we prove the chosen-plaintext attack (CPA) security of the fully homomorphic scheme and the circular security of key switching process in standard model under the learning with errors problem (LWE) assumption.
Abstract. In this paper we port Brakerski’s fully homomorphic scheme based on the Learning With Errors (LWE) problem to the ring-LWE setting. We introduce two optimised versions of relinearisation that not only result in a smaller relinearisation key, but also faster computations. We provide a detailed, but simple analysis of the various homomorphic operations, such as multiplication, relinearisation and bootstrapping, and derive tight worst case bounds on the noise caused by these operations. The analysis of the bootstrapping step is greatly simplified by using a modulus switching trick. Finally, we derive concrete parameters for which the scheme provides a given level of security and becomes fully homomorphic. 1
Abstract In this paper, we revisit the fully homomorphic encryption (FHE) scheme implemented by Gentry and Halevi, which is just an instantiation of Gentry's original scheme based on ideal lattices. Their FHE scheme starts from a somewhat homomorphic encryption (SHE) scheme, and its decryption range is deeply related with the FHE construction. Gentry and Halevi gave an experimental evaluation of the decryption range, but theoretical evaluations have not been given so far. Moreover, we give a theoretical upper bound, and reconsider suitable parameters for theoretically obtaining an FHE scheme. In particular, while Gentry and Halevi use the Euclidean norm evaluation in the noise management of ciphertexts, our theoretical bound enables us to use the ∞-norm evaluation, and hence it helps to lower the difficulty of controlling the noise density of ciphertexts.