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A Latticed-Based Public Key Encryption with KDM Security from R-LWE

Yan Fang Wu, Zheng Huang, Qiao Yan Wen

Published Sep 1, 2012• 0 citations
Publisher

Abstract

Since the introduction of the ring learning with errors (R-LWE) by Lyubashevsky, Peikert and Regev, many efficient and secure applications were founded in cryptography. In this paper, we mainly present an efficient public-key encryption scheme based on the R-LWE assumption. It is very simple to describe and analyze. As well as it can achieve security against certain key-dependent message (KDM) attacks. Namely, this efficient encryption scheme can securely encrypt its own secret key. The security of this scheme follows from the already proven hardness of the R-LWE problem since the R-LWE assumption is reducible to worst-case problems on the ideal lattice. Besides, the scheme enjoys a high level efficiency and low cost since the operations of the scheme are very simple and fast. The cost of both the encryption and decryption is only polylog(n) bit per message symbol.

DOI
10.4028/www.scientific.net/aef.6-7.398

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Since the introduction of the ring learning with errors (R-LWE) by Lyubashevsky, Peikert and Regev, many efficient and secure applications were founded in cryptography. In this paper, we mainly present an efficient public-key encryption scheme based on the R-LWE assumption. It is very simple to describe and analyze. As well as it can achieve security against certain key-dependent message (KDM) attacks. Namely, this efficient encryption scheme can securely encrypt its own secret key. The security of this scheme follows from the already proven hardness of the R-LWE problem since the R-LWE assumption is reducible to worst-case problems on the ideal lattice. Besides, the scheme enjoys a high level efficiency and low cost since the operations of the scheme are very simple and fast. The cost of both the encryption and decryption is only polylog(n) bit per message symbol.
Key-dependent message (KDM) security is of great research significance, to better analyse and solve the potential security problems in complex application scenarios. Most of the current KDM security schemes are based on traditional hard mathematical problems, where the public key and ciphertext are not compact enough, and make the ciphertext size grow linearly with the degree of the challenge functions. To solve the above problems and the inefficient ciphertext operation, the authors propose a compact lattice-based cryptosystem with a variant of the RLWE problem, which applies an invertible technique to obtain the RLWE* problem. It remains hard after the modification from the RLWE problem. Compared with the ACPS scheme, our scheme further expands the set of challenge functions based on the affine function of the secret key, and the size of public key and ciphertext is O˜(n), which is independent of the challenge functions. In addition, this scheme enjoys a high level of efficiency, the cost of encryption and decryption is only ploylog(n) bit operations per message symbol, and we also prove that our scheme is KDM-CPA secure under the RLWE* assumption.
Abstract. We construct the first public-key encryption scheme that is proven secure (in the standard model, under standard assumptions) even when the attacker gets access to encryptions of arbitrary efficient functions of the secret key. Specifically, under either the DDH or LWE assumption, and for arbitrary but fixed polynomials L and N, we obtain a public-key encryption scheme that resists key-dependent message (KDM) attacks for up to N(k) public keys and functions of circuit size up to L(k), where k denotes the size of the secret key. We call such a scheme bounded KDM secure. Moreover, we show that our scheme suffices for one of the important applications of KDM security: ability to securely instantiate symbolic protocols with axiomatic proofs of security. We also observe that any fully homomorphic encryption scheme that additionally enjoys circular security and circuit privacy is fully KDM secure in the sense that its algorithms can be independent of the polynomials L and N as above. Thus, the recent fully homomorphic encryption scheme of Gentry (STOC 2009) is fully KDM secure under certain non-standard hardness assumptions. Finally, we extend an impossibility result of Haitner and Holenstein (TCC 2009), showing that it is impossible to prove KDM security against a family of query functions that contains exponentially hard pseudo-random functions if the proof makes only a black-box use of the query function and the adversary attacking the scheme. This shows that the non-black-box use of the query function in our proof of security is inherent.
We propose a variant version of ring learning with errors (R-LWE) assumption. Under the modified slightly assumption which is reducible to the worst-case problems on ideal lattice, we present a construction of digital signatures. So the scheme is provably secure based on the hardness of lattice problems (such as approximating the length of the shortest vector within a fixed poly(n) factor). Compared with some existing typical lattice-based signature schemes, the construction enjoys many advantages, including simple and efficient key generation algorithm, signing algorithm and verification algorithm to improve the efficiency of the scheme as well as to reduce the overhead. The sizes of the secret and public keys and the signature are almost linear O(n log n) (up to poly-logarithmic factors) in the dimension n of the lattice.
Attribute-based encryption (ABE) has been an active research area in cryptography due to its attractive applications. But almost all attribute-based encryption schemes are based on bilinear maps, which leave them vulnerable to quantum cryptanalysis. The lattice-based ABE schemes from the Learning with errors (LWE) have appeared, but they are not efficient enough for practical applications. Thus we propose an efficient attribute-based encryption based on the Learning with errors over Rings (R-LWE), which is called ABER-LWE. The security analysis shows that ABER-LWEscheme is secure in the selective-set model under the R-LWE assumption, whose security can reduce to the hardness of the shortest vector problem in the worst case on ideal lattices. The efficiency analysis indicates that ABER-LWEis more efficient than previous ABE cryptosystems on lattices.
Based on the learning with errors (LWE) problem, we construct an identity-based encryption (IBE) scheme in the standard-model which is secure against the key dependent message (KDM) attacks and the selective opening (SO) attacks. KDM security, which requires that the scheme can still obtain secrecy even if the messages depend on the secret key, is necessary for IBE when updating the identity secret keys with the old ones. SO security, which requires that when the adversary corrupts some fraction of senders, the uncorrupted senders' messages still maintain privacy, is useful for IBE since that one user receives messages from many senders is a natural scenario. Although there is a KDM secure IBE scheme based on LWE constructed by Alperin-Sheriff and Peikert (PKC'12), and a SO secure IBE scheme based on LWE constructed by He et al. (IWSEC'15), those two schemes cannot achieve both of the security requirements at the same time.
Compared to the classical cryptography, lattice-based cryptography is more secure, flexible and simple, and it is believed to be secure against quantum computers.In this paper, an efficient signature scheme is proposed from the ring learning with errors (R-LWE), which avoids sampling from discrete Gaussians and has the characteristics of the much simpler description etc.Then, the scheme is implemented in C/C++ and makes a comparison with the RSA signature scheme in detail.Additionally, a linearly homomorphic signature scheme without trapdoor is proposed from the R-LWE assumption.The security of the above two schemes are reducible to the worst-case hardness of shortest vectors on ideal lattices.The security analyses indicate the proposed schemes are unforgeable under chosen message attack model, and the efficiency analyses also show that the above schemes are much more efficient than other correlative signature schemes.