Lattice-based Public-key Encryption Research Articles

A lattice-based public key encryption scheme with delegated equality test

When data security is facing growing threats, ordinary encryption techniques cannot meet the needs of comparing, sharing, and classifying data hidden in ciphertexts. At the same time, the advent of the quantum computing era has brought unprecedented challenges to traditional cryptography. Fortunately, a lattice-based PKEET scheme can solve the above problems. In this paper, we design a lattice-based PKE-DET scheme that can support the delegated tester function with satisfying anti-quantum computing security and resisting OMRA attacks. To the best of our knowledge, this is the first PKE-DET scheme that has both kinds of security at the same time. Under the standard model, we prove the security of the scheme based on the LWE hardness assumption. Compared with existing schemes, our scheme has many advantages, such as high security, delegated tester authorization, and small storage space.

Inner product encryption from Middle-Product Learning With Errors

Inner product encryption (IPE) is an important research area of functional cryptosystems. It can improve user access control and fine-grained query, and has a wide range of applications in emerging fields such as cloud computing. Lattice-based inner product encryption has the advantages of resistance to quantum algorithm attacks and relatively simple encryption algorithms, and thus has good application prospects. Currently, the provable security of lattice-based public key encryption schemes is mostly based on the learning with errors (LWE) and polynomial learning with errors (PLWE) problems. However, most of the encryption schemes based on LWE problems suffer from the problem of oversized public keys and ciphers. Although encryption schemes based on the PLWE problem further reduce the size, their hardness is limited by polynomials and the security guarantees are weakened. Compared with the LWE and PLWE problems, the Middle-Product Learning With Errors (MP-LWE) problem relaxes the polynomial restriction and makes full use of the polynomial features to achieve a better balance between security and efficiency. Inspired by this, we have proposed the inner product encryption scheme of Sel-IND-CPA-secure in SocialSec 2022. In this paper, to extend the existing work and improve the functionality of IPE, we construct single-input and multi-input inner product encryption schemes of Ad-IND-CPA-secure, and also evaluate the efficiency of the schemes.

On Advances of Lattice-Based Cryptographic Schemes and Their Implementations

Lattice-based cryptography is centered around the hardness of problems on lattices. A lattice is a grid of points that stretches to infinity. With the development of quantum computers, existing cryptographic schemes are at risk because the underlying mathematical problems can, in theory, be easily solved by quantum computers. Since lattice-based mathematical problems are hard to be solved even by quantum computers, lattice-based cryptography is a promising foundation for future cryptographic schemes. In this paper, we focus on lattice-based public-key encryption schemes. This survey presents the current status of the lattice-based public-key encryption schemes and discusses the existing implementations. Our main focus is the learning with errors problem (LWE problem) and its implementations. In this paper, the plain lattice implementations and variants with special algebraic structures such as ring-based variants are discussed. Additionally, we describe a class of lattice-based functions called lattice trapdoors and their applications.

Quantum Misuse Attack on Frodo

Research on the security of lattice-based public-key encryption schemes against misuse attacks is an important part of the cryptographic assessment of the National Institute of Standards and Technology (NIST) post-quantum cryptography (PQC) standardization process. In particular, many NIST-PQC cryptosystems follow the same meta-cryptosystem. At EUROCRYPT 2019, Betu et al. mounted a classical key recovery under plaintext checking attacks (KR-PCA) and a quantum key recovery under chosen ciphertext attacks (KR-CCA). They analyzed the security of the weak version of nine submissions to NIST. In this paper, we focus on learning with error (LWE)-based FrodoPKE, whose IND-CPA security is tightly related to the hardness of plain LWE problems. We first review the meta-cryptosystem and quantum algorithm for solving quantum LWE problems. Then, we consider the case where the noise follows a discrete Gaussian distribution and recompute the success probability for quantum LWE by using Hoeffding bound. Finally, we give a quantum key recovery algorithm based on LWE under CCA attack and analyze the security of Frodo. Compared with the existing work of Betu et al., our method reduces the number of queries from to 1 with the same success probability.

Lattice-based public-key encryption with equality test supporting flexible authorization in standard model

As cloud computing has developed rapidly, outsourcing data to cloud servers for remote storage has become an attractive trend. However, when cloud clients store their data in the cloud, the security and privacy of cloud data would be threatened due to accidental corruptions or purposive attacks caused by a semi-trusted cloud server. The widely used method of addressing the security and privacy of cloud data is to store encrypted data instead of plain data. As the resulting system is unusable, since the cloud can no longer search throughout the data, new cryptographic primitive such as public-key encryption with equality test (PKEET) has been introduced. PKEET has many interesting applications such as keyword search on encrypted data, encrypted data partitioning for efficient encrypted data management, personal health record systems, spam filtering in encrypted email systems and so forth. PKEET allows checking whether two ciphertexts encrypted under different public keys contain the same message or not. However, unrestricted access to equality tests can reveal information about the underlying data. This is not acceptable in respect of users' privacy. In 2015, Ma et al. introduce the notion of PKEET with flexible authorization (PKEET-FA) which strengthens privacy protection. Since 2015, there are several follow-up works on PKEET-FA. However, all are secure in the random oracle model and vulnerable to quantum attacks. In this paper, we provide two constructions of quantum-safe PKEET-FA secure in the standard model based on the hardness assumptions of integer lattices and ideal lattices. Finally, we implement the PKEET-FA scheme over ideal lattices.

Lattice-based public-key encryption with conjunctive keyword search in multi-user setting for IIoT

Lattice-based public-key encryption with conjunctive keyword search in multi-user setting for IIoT

Lattice-Based Public Key Encryption with Multi-Ciphertexts Equality Test in Cloud Computing

Lattice-Based Public Key Encryption with Multi-Ciphertexts Equality Test in Cloud Computing

On Exploiting Message Leakage in (Few) NIST PQC Candidates for Practical Message Recovery Attacks

In this work, we propose generic and practical side-channel attacks for message recovery in post-quantum lattice-based public key encryption (PKE) and key encapsulation mechanisms (KEM). The targeted schemes are based on the well known Learning With Errors (LWE) and Learning With Rounding (LWR) problem and include three finalists and six semi-finalist candidates of the ongoing NIST’s standardization process for post-quantum cryptography. Notably, we propose to exploit inherent <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ciphertext malleability</i> properties of LWE/LWR-based PKEs as a powerful tool for side-channel assisted message recovery attacks. The use of ciphertext malleability widens the scope of previous attacks with the ability to target multiple operations for message recovery. Moreover, our attacks are adaptable to different implementation variants and are also applicable to implementations protected with concrete <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">shuffling</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">masking</i> side-channel countermeasures. Our work mainly highlights the presence of inherent algorithmic properties in LWE/LWR-based schemes that can aid side-channel attacks for message recovery, thereby stressing on the need for strong side-channel countermeasures against message recovery for LWE/LWR-based schemes.

Chosen-ciphertext lattice-based public key encryption with equality test in standard model

With the rapid growth of cloud storage and cloud computing services, many organizations and users choose to store the data on a cloud server for saving costs. However, due to security concerns, data of users would be encrypted before sending to the cloud. However, this hinders a problem of computation on encrypted data in the cloud, especially in the case of performing data matching in various medical scenarios. Public key encryption with equality test (PKEET) is a powerful tool that allows the authorized cloud server to check whether two ciphertexts are generated by the same message. PKEET has then become a promising candidate for many practical applications like efficient data management on encrypted databases. Lee et al. (Information Sciences 2020) proposed a generic construction of PKEET schemes in the standard model and hence it is possible to yield the first instantiation of post-quantum PKEET schemes based on lattices. At ACISP 2019, Duong et al. proposed a direct construction of PKEET over integer lattices in the standard model. However, their scheme does not reach the CCA2-security. In this paper, we propose an efficient CCA2-secure PKEET scheme based on ideal lattices. In addition, we present a modification of the scheme by Duong et al. over integer lattices to attain the CCA2-security. Both schemes are proven secure in the standard model, and they enjoy the security in the upcoming quantum computer era.

Logic Programming with Post-Quantum Cryptographic Primitives for Smart Contract on Quantum-Secured Blockchain.

This paper investigates the usage of logic and logic programming in the design of smart contracts. Our starting point is the logic-based programming language for smart contracts used in a recently proposed framework of quantum-secured blockchain, called Logicontract (LC). We then extend the logic used in LC by answer set programming (ASP), a modern approach to declarative logic programming. Using ASP enables us to write various interesting smart contracts, such as conditional payment, commitment, multi-party lottery and legal service. A striking feature of our ASP implementation proposal is that it involves post-quantum cryptographic primitives, such as the lattice-based public key encryption and signature. The adoption of the post-quantum cryptographic signature overcomes a specific limitation of LC in which the unconditionally secure signature, despite its strength, offers limited protection for users of the same node.

A public key encryption scheme based on a new variant of LWE with small cipher size

The lattice cryptosystem is considered to be able to resist the attacks of quantum computers. Lattice-based Public Key Encryption (PKE) schemes have attracted the interest of many researchers. In lattice-based cryptography, Learning With Errors (LWE) problem is a hard problem usually used to construct PKE scheme. To ensure the correctness of decryption, LWE-based schemes have a large ciphertext size. This makes these encryption schemes not practical enough when the communication bandwidth is limited. We propose a new variant of LWE, named Learning With Modulus (LWM) and prove that the new problem can be reduced from LWE problem. The proof idea of our reduction is similar to the reduction of LWR problem. We also construct a new PKE scheme based on the proposed LWM and LWE, which has small ciphertext size. For a 128 bits plaintext, the ciphertext size of our scheme is 53.57% of Lindner–Peikert’s (LP) scheme under the same security level. We use python to test the performance of our scheme. The results show that our scheme is only about 0.015 ms slower than LP in the decryption. The performance of our scheme for generating keys and encrypting messages is similar to LP.

Post-quantum secure multi-party private set-intersection in star network topology

In many realistic scenarios, participants wish to perform some secret set operations such as intersection, union, cardinality of intersection, etc. on their private data sets. Private Set Intersection (PSI) plays a major role in addressing such problems. PSI is one of the widely used secure multi-party computation technique that allows the participants to securely compute the intersection of their private input sets and nothing beyond that. It is generally executed between two parties. When the number of entities is more than two, it is known as multi-party PSI (MPSI). Today, the security of all the existing MPSI protocols are based on number theoretic assumptions. However, these will become insecure once large enough quantum computers are built. As a consequence, designing of quantum computer resistant MPSI becomes an interesting direction of research work. This paper addresses the issue by presenting the first post-quantum MPSI protocol in the so-called star network topology, using lattice-based public key encryption scheme. We utilize space-efficient probabilistic data structure (Bloom filter) as building blocks of our design. It attains security in standard model (without random oracles) under the decisional learning with errors (DLWE) assumption.

Batten Down the Hatches: Securing Neighborhood Area Networks of Smart Grid in the Quantum Era

On account of the recent advances in quantum computers, it becomes pressing to consider quantum-safe authentication schemes for two-way communications in neighborhood area networks of smart grid, i.e., preventing possible attacks that the quantum computers could bring to the grid. In this paper, we take a further step toward this goal by integrating the lattice-based public-key encryption and key exchange techniques to provide mutual authentication between smart meters and the neighborhood gateway. Security analysis shows that our proposed scheme is secure under attacks from both quantum and conventional computers, including the recently introduced key reuse attacks. In addition, the proposed schemes can also achieve forward security, which guarantees that the secrets leaked in the current session will not result in the leakage of secrets in former sessions. Furthermore, the proposed scheme also gives prior consideration to the smart meters with limited computation capacity and puts as little burden on them as possible. Extensive performance analysis is also conducted, and the results demonstrate the efficiency of our proposed scheme, as well as the relatively low communication and storage costs.

Design and Implementation of a Lattice-Based Public-Key Encryption Scheme

Due to its advantage of quantum resistance and the provable security under some worst-case hardness assumptions, lattice-based cryptography is being increasingly researched. This paper tries to explore and present a novel lattice-based public key cryptography and its implementation of circuits. In this paper, the LWE (learning with error) cryptography is designed for circuit realization in a practical way. A strategy is proposed to dramatically reduce the stored public key size from [Formula: see text] to [Formula: see text], with only several additional linear feedback shift registers. The circuit design is implemented on Xilinx Spartan-3A FPGA and performs very well with limited resources. Only 125 slices and 8 BRAMs are occupied, and there are no complex operation devices such as multipliers or dividers, all the involved arithmetic operations are additions. This design is smaller than most hardware implementations of LWE or Ring-LWE cryptography in current state, while having an acceptable frequency at 111 MHz. Therefore, LWE cryptography can be practically realized, and its advantages of quantum resistance and simple implementation make the public key cryptography promising for some applications in devices such as smart cards.

A Lattice-Based Group Authentication Scheme

Authentication has been adopted in many areas, but most of these authentication schemes are built using traditional cryptographic primitives. It is widely believed that such primitives are not resistant to quantum algorithms. To deal with those quantum attacks, lattice-based cryptography was introduced by Ajtai in 1996. To the best of our knowledge, the existing lattice-based authentication schemes are based on a lattice-based public key encryption called NTRU: a ring-based public key cryptosystem, proposed by Hoffstein, Pipher, and Silverman in 1998. However, these schemes only support the case of a single user. In view of the aforementioned issue, we propose the first lattice-based group authentication scheme. The proposed scheme is secure against replay attacks and man-in-the-middle attacks. Moreover, compared with the existing lattice-based authentication schemes, ours provides the most efficient method to agree upon a session key among a group of users after mutual authentication.

An Efficient Public Key Encryption Scheme Based on QC-MDPC Lattices

In this paper, we introduce a new family of lattices, namely QC-MDPC lattices, which are a special case of LDPC lattices, and an improved bit flipping algorithm for decoding of these lattices. Encoding and decoding implementations of QC-MDPC lattices are practical in high dimensions. Indeed, to take advantage of practical decoding, we use “Construction-A” lattices which makes a tight connection between the structure of lattices and codes. Using these features, we design a lattice-based public key encryption scheme enjoying linear encryption and decryption complexities. The proposed scheme has a reasonable key size due to the sparseness of the parity-check matrix, and the quasi-cyclic structure of the parity-check and generator matrices. Besides, the message expansion of the proposed scheme is smaller than other lattice-based and code-based cryptosystems with comparative parameters. All these features provide a lattice-based public key encryption scheme with reasonable key size, linear encryption, and decryption algorithms and small message expansion. On the other hand, we show that the cryptosystem is resistant against all known attacks both on lattice-based and code-based cryptosystems for different levels of security.

Packing Messages and Optimizing Bootstrapping in GSW-FHE

We construct the first fully homomorphic encryption (FHE) scheme that encrypts matrices and supports homomorphic matrix addition and multiplication. This is a natural extension of packed FHE and thus supports more complicated homomorphic operations. We optimize the bootstrapping procedure of Alperin-Sheriff and Peikert (CRYPTO 2014) by applying our scheme. Our optimization decreases the lattice approximation factor from \(\tilde{O}(n^{3})\) to \(\tilde{O}(n^{2.5})\). By taking a lattice dimension as a larger polynomial in a security parameter, we can also obtain the same approximation factor as the best known one of standard lattice-based public-key encryption without successive dimension-modulus reduction, which was essential for achieving the best factor in prior works on bootstrapping of standard lattice-based FHE.

Chosen ciphertext attacks on lattice-based public key encryption and modern (non-quantum) cryptography in a quantum environment

Modern cryptography is based on various building blocks such as one way functions with or without trapdoors, pseudo-random functions, one way permutations with or without trapdoors, etc. In a quantum world some of the main candidates for these building blocks are broken. For instance, the security of the most popular public-key cryptosystem—RSA—is related to the difficulty of factoring large numbers, and is broken (in principle) by a quantum computer. We investigate some of the remaining candidates, and discuss the resulting “Post-Quantum Cryptography” (namely, the resulting “modern cryptography in a quantum environment”). About half a decade ago Ajtai and Dwork (and later on, also Goldreich, Goldwasser and Halevi) proposed a public key cryptosystem that has a proven security under a plausible complexity assumption. The plausible assumption is that the so-called unique shortest vector problem (u-SVP) is hard on the worst case. This problem is potentially still hard also in a quantum environment. Recently, Regev introduced a new (and much simpler) public key cryptosystem, based on the same u-SVP hardness assumption, but with improved parameters. In this paper we present chosen ciphertext attacks (CCA) against all three cryptosystems. Our attack shows that these cryptosystems are totally insecure against CCA, because the private keys can be recovered in polynomial time. We then discuss the possibility of making public key encryption (PKE) secure against CCA, without adding stronger assumptions than the assumption that u-SVP is hard. We conclude that the current understanding of modern cryptography in a quantum environment can only suggest CCA-secure interactive-PKE, which is obviously weaker than CCA-secure PKE. Finally, we discuss the relation of our attack to the reaction attack of Hall, Goldberg and Schneier, which we only recently became aware of.