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Identity-Based Matrix (Level) Fully Homomorphic Encryption Scheme

CHEN Hong, HUANG Jie, CHEN Honglin, WANG Runting, XIAO Chenglong, GUO Pengfei, JIN Haibo

Published Oct 1, 2020• 1 citation

Abstract

With the development of quantum theory, traditional encryption algorithms are facing enormous challenges. Once quantum computers are available, existing encryption systems will no longer be secure. The lattice cipher is proven by theory to be a new type of encryption scheme, which can resist quantum algorithm. It has the property of fast computing speed and small space occupation, hence, it has become a hot issue in today??s research. The identity-based encryption scheme can improve the utilization of storage space and facilitate user public key management. Consequently, this paper based on the hard problem of learning with errors (LWE), constructs an identity-based matrix encryption scheme at first, then applies the improved key switching technology to reduce the dimension of homomorphic ciphertext to normal, and uses the mode conversion technique to reduce noise, realizing matrix homo-morphic operation, and getting an identity-based matrix (level) fully homomorphic encryption scheme. The scheme uses matrices as the storage structure, which has greatly improved the processing capability compared to other single-bit schemes. The scheme not only satisfies the addition and Hadamard product of the matrix, but also supports matrix dot multiplication. The experimental results show that compared with other methods, the improved key switching technology in this paper inputs less noise. Through the difficulty of the LWE problem, it is proven that the scheme satisfies IND-sID-CPA (indistinguishability-selective identity-chosen plaintext attack).

DOI: 10.3778/j.issn.1673-9418.1910061

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With the development of quantum theory, traditional encryption algorithms are facing enormous challenges. Once quantum computers are available, existing encryption systems will no longer be secure. The lattice cipher is proven by theory to be a new type of encryption scheme, which can resist quantum algorithm. It has the property of fast computing speed and small space occupation, hence, it has become a hot issue in today??s research. The identity-based encryption scheme can improve the utilization of storage space and facilitate user public key management. Consequently, this paper based on the hard problem of learning with errors (LWE), constructs an identity-based matrix encryption scheme at first, then applies the improved key switching technology to reduce the dimension of homomorphic ciphertext to normal, and uses the mode conversion technique to reduce noise, realizing matrix homo-morphic operation, and getting an identity-based matrix (level) fully homomorphic encryption scheme. The scheme uses matrices as the storage structure, which has greatly improved the processing capability compared to other single-bit schemes. The scheme not only satisfies the addition and Hadamard product of the matrix, but also supports matrix dot multiplication. The experimental results show that compared with other methods, the improved key switching technology in this paper inputs less noise. Through the difficulty of the LWE problem, it is proven that the scheme satisfies IND-sID-CPA (indistinguishability-selective identity-chosen plaintext attack).

Due to the threat of future quantum computers, it is necessary to develop cryptographic schemes that are resistant to quantum attacks. Lattice based cryptographic schemes have resistance to quantum attacks and security under worst-case assumptions that are difficult to solve. To solve the problems of large size of public and private keys, low encryption efficiency, and large noise growth in identity based encryption schemes, an adaptive secure identity based all homomorphic encryption scheme based on Learning with Errors (LWE) is constructed. This scheme uses {0,1} matrices to store plaintext, significantly improving encryption efficiency compared to single bit encryption schemes. It uses key conversion technology to achieve homomorphic operations and modulus conversion technology to reduce noise. It also converts IDs into bit sequences to construct an encryption scheme for adaptive security. Compared to selective security that only provides security guarantees for specific types of attacks, adaptive security is suitable for general scenarios, providing higher security guarantees and being able to cope with more challenging attack situations. It has been proven through the LWE problem that the security of our proposed scheme satisfies IND-ID-CPA. Finally, the experimental results indicate that this scheme introduces less noise, and has higher encryption efficiency and performance.

The lattice-based multi-identity fully homomorphic encryption scheme combines the quantum security of lattice cryptography with the advantage of identity-based encryption. However, existing schemes face challenges such as large key sizes, inefficient ciphertext expansion processes, and reliance on outdated trapdoor designs, limiting their compactness and practicality. In this study, we propose a novel Compact Multi-Identity Fully Homomorphic Encryption Scheme (WZ-MIBFHE) that eliminates the need for fresh ciphertexts during expansion. First, we construct a compact identity-based encryption scheme by combining the YJW23 trapdoor and ABB10 under the standard model, proving its IND-sID-CPA security. The scheme is then adapted to ensure correctness and security when integrated with the decomposition method for ciphertext expansion. This adaptation also utilizes approximation errors to reduce overall noise. Finally, we expand the modified IBE scheme’s ciphertext using the decomposition method to construct the WZ-MIBFHE scheme. Compared to existing methods, WZ-MIBFHE reduces the lattice dimension to nlogq+logbq, improves public and private key sizes, and significantly lowers ciphertext expansion rates by removing the need for fresh ciphertexts. These improvements enhance both the compactness and efficiency of the scheme, making it a promising solution for multi-identity homomorphic encryption.

1 (State Key Laboratory of Mathematical Engineering and Advanced Computing , Zhengzhou 450001) Abstract: Fully homomorphic is an encryption scheme that allows for data to be stored and processed in an encrypted format, which gives the cloud provider a solution to host and process data without even knowing what the message is. In previous identity-based homomorphic encryption scheme, computing efficiency is complicated and expensive. In this work, based on Regev’s work, we propose a sampling trapdoor one-way function in arbitrary cyclotomic rings [ ] ( ) / 1 n R X x = + Z . Then construct a leveled identity-based homomorphic encryption scheme from ring learning with errors, which has advantage in computational efficiency and key management, by using user’s identity as the unique public key. This scheme is proved IND-CPA secure in the random oracle model, relied to hardness of decision ring learning with errors problem.

This paper proposes an efficient identity-based leveled fully homomorphic encryption (IBFHE) scheme over ideal lattice. First, we achieve an identity-based encryption (IBE) scheme in dual-LWE cryptosystem. Then, we take advantage of the eigenvector method to compile the IBE scheme into an IBFHE scheme. Compared with the previous scheme, our scheme is of shorter key size, smaller storage, and higher efficiency, owing to a new trapdoor generation function and algebraic structure of ideal lattice. The security analysis shows that our scheme is selective-identity secure against chosen-plaintext attacks in the standard model.

Abstract Based on the learning with errors over rings (RLWE) assumption, a leveled fully homomorphic encryption (FHE) by the approximate eigenvector method under the same public key is proposed, which security can be reduced to the shortest vector problem on ideal lattices in the worst case. And the leveled FHE under different public keys is realized by the secret key switching without dimension reduction. Combine the public key of leveled FHE with the identity, we bring forward an efficient identity‐based leveled FHE scheme from the RLWE assumption. The security analysis shows that our identity‐based leveled FHE scheme is selective‐ID secure against chosen‐plaintext attacks in the random oracle model. And the efficiency analysis and simulation results show that the proposed identity‐based leveled FHE scheme is much more efficient than Gentry's and Wang's identity‐based cryptosystems based on the learning with errors assumption. Copyright © 2016 John Wiley & Sons, Ltd.

An efficient identity-based encryption (IBE) scheme over lattice is proposed in this paper. Under the hardness of the learning with errors (LWE) problem, the proposed scheme is semantic secure against adaptive chosen identity and chosen plaintext attack in the standard model. To improve the efficiency of the lattice-based IBE scheme, unlike the identity string is encoded into a matrix by a group of public matrices in several known constructions, the identity string of l bits is encoded into a vector with the help of l+1 vectors in this paper. With the help of this idea, we achieve the private key extraction of IBE scheme at the same lattice. Then, the public key of the proposed scheme only consists of one n×m matrix and l+1 vectors, compared with that the public keys of the known lattice-based IBE schemes all consist as a group of n×m matrices. Hence, the public key size of this scheme is shorter than that of the known constructions.