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Spectral distortion and the ring learning with errors problem

INVOLVE, A JOURNAL OF MATHEMATICS 2025年第2期18卷 181-198页

作者： Babinkostova, Liljana Chin, Ariana G. Kirtland, Aaron Nazarchuk, Vladyslav Plotnick, Esther Boise State Univ Dept Math Boise ID 83725 USA Univ Calif Los Angeles Dept Math Los Angeles CA USA Washington Univ St Louis St Louis MO USA Yale Univ New Haven CT USA Harvard Univ Cambridge MA USA

Certain attacks on the ring learning with errors (RLWE) problem rely on reduction to the polynomial variant (PLWE) of the problem. The spectral distortion is used to measure the RLWE-to-PLWE reduction. We investigate the spectral distortion and answer questions posed by Y. Elias et al. (2016) regarding spectral distortion used as a condition for a successful attack. We provide a closed form for the spectral distortion of certain classes of cyclotomic polynomials. Moreover, we present a bound on the spectral distortion of cyclotomic polynomials.

关键词： learning with errors spectral distortion cyclotomic polynomials

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Rényi divergence on learning with errors

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Science China(Information Sciences) 2020年第9期63卷 124-139页

作者： Yang TAO Han WANG Rui ZHANG State Key Laboratory of Information Security Institute of Information EngineeringChinese Academy of Sciences School of Cyber Security University of Chinese Academy of Sciences

Many lattice-based schemes are built from the hardness of the learning with errors problem, which naturally comes in two flavors: the decision LWE and search LWE. In this paper, we investigate the decision LWE and search LWE by Rényi divergence respectively and obtain the following results: For decision LWE,we apply RD on LWE variants with different error distributions(i.e., center binomial distribution and uniform distribution, which are frequently used in the NIST PQC submissions) and prove the pseudorandomness in theory. As a by-product, we extend the so-called public sampleability property and present an adaptively public sampling property to the application of Rényi divergence on more decision problems. As for search LWE, we improve the classical reduction proof from Gap SVP to LWE. Besides, as an independent interest,we also explore the intrinsic relation between the decision problem and search problem.

关键词： Rényi divergence learning with errors decision problems search problems post-quantum cryptography

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COLLUSION RESISTANT TRAITOR TRACING FROM learning with errors

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SIAM JOURNAL ON COMPUTING 2020年第5期49卷 STOC18-94-STOC18-241页

作者： Goyal, Rishab Koppula, Venkata Waters, Brent Univ Texas Austin Dept Comp Sci Austin TX 78712 USA

In this work we provide a traitor tracing construction with ciphertexts that grow polynomially in log(n), where n is the number of users, and prove it secure under the learning with errors (LWE) assumption. This is the first traitor tracing scheme with such parameters provably secure from a standard assumption. In addition to achieving new traitor tracing results, we believe our techniques push forward the broader area of computing on encrypted data under standard assumptions. Notably, traitor tracing is a substantially different problem from other cryptography primitives that have seen recent progress in LWE solutions. We achieve our results by first conceiving a novel approach to building traitor tracing that starts with a new form of functional encryption that we call Mixed FE. In a Mixed FE system the encryption algorithm is bimodal and works with either a public key or master secret key. Ciphertexts encrypted using the public key can only encrypt one type of functionality. On the other hand, the secret key encryption can be used to encode many different types of programs, but is only secure as long as the attacker sees a bounded number of such ciphertexts. We first show how to combine mixed FE with attribute-based encryption to achieve traitor tracing. Second, we build Mixed FE systems for polynomial-sized branching programs (which corresponds to the complexity class logspace) by relying on the polynomial hardness of the LWE assumption with superpolynomial modulus-to-noise ratio.

关键词： traitor tracing private linear broadcast encryption functional encryption Mixed FE attribute-based encryption lattice trapdoors encryption decryption learning with errors

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Leveled Certificateless Fully Homomorphic Encryption Schemes From learning with errors

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IEEE ACCESS 2020年 8卷 26749-26763页

作者： Li, Mingxiang Hebei Finance Univ Inst Financial Res Baoding 071051 Peoples R China Sci & Technol Finance Key Lab Hebei Prov Baoding 071051 Peoples R China

Fully homomorphic encryption (FHE) is a form of public-key encryption that allows the computation of arbitrary functions on encrypted data without decrypting the data. As a result, it is a useful tool with numerous applications. Certificateless encryption (CLE) is a type of public-key encryption that combines the advantages of PKI-based public-key encryption with those of identity-based encryption (IBE). Thus, certificateless fully homomorphic encryption (CLFHE) has aroused considerable research interest. Recently, Chen, Hu, and Lian proposed a leveled certificateless homomorphic encryption (CLHE) scheme and proved its semantic security based on the learning with errors (LWE) problem in the random oracle model. However, their scheme supports only homomorphic addition, but not homomorphic multiplication. In this work, we construct two leveled CLFHE schemes using the approximate eigenvector method presented by Gentry, Sahai, and Waters. Based on the hardness of the LWE problem, we prove that one scheme satisfies adaptive semantic security and anonymity in the random oracle model, whereas the other satisfies selective semantic security and anonymity in the standard model.

关键词： Certificateless fully homomorphic encryption learning with errors random oracle model standard model

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Delegation of Decryption Rights With Revocability From learning with errors

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IEEE ACCESS 2018年 6卷 61163-61175页

作者： Yin, Wei Wen, Qiaoyan Liang, Kaitai Zhang, Zhenfei Chen, Liqun Yan, Hanbing Zhang, Hua Beijing Univ Posts & Telecommun State Key Lab Networking & Switching Technol Beijing 100876 Peoples R China Univ Surrey Guildford GU2 7XH Surrey England Onboard Secur Inc Wilmington MA 01887 USA Coordinat Ctr China Natl Comp Network Emergency Response Tech Team Beijing 100020 Peoples R China

The notion of decryption rights delegation was initially introduced by Blaze et al. in EURO-CRYPT 1998. It, defined as proxy re-encryption, allows a semi-trusted proxy to convert a ciphertext intended for a party to another ciphertext of the same plaintext, without knowledge of the underlying plaintext and decryption key. It has been explored to many real-world applications, e.g., encrypted email forwarding. However, the intrinsic all-or-nothing share feature of proxy re-encryption yields a limitation that the share cannot be revoked. This may hinder the scalability of its applications in practice. In this paper, for the first time, we define the concept of revocability in terms of decryption rights delegation. The novel concept enables data owner to revoke the shared decryption rights when needed. Inspired by the seminal lattice-based proxy re-encryption proposed in PKC 2014, we design a concrete lattice-based construction which satisfies the notion. In our construction, we make use of binary-tree structure to implement the revocation of decryption rights, so that the update of re-encryption key is reduced to O(logN) (instead of O(N)), where N is the maximum number of delegatee. Furthermore, the security of our scheme is based on the standard learning with errors problem (LWE problem), which could be reduced to the worst-case hard problems (such as GapSVP and SIVP) in the context of lattices. The scheme is chosen ciphertext secure in the standard model. As of independent interest, our scheme achieves both backward and forward security, which means that once a user is revoked after a time period t, it cannot gain access to all encrypted files before and after t.

关键词： Revocability proxy re-encryption lattice learning with errors

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Comparison of encrypted control approaches and tutorial on dynamic systems using learning with errors-based homomorphic encryption

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ANNUAL REVIEWS IN CONTROL 2022年 54卷 200-218页

作者： Kim, Junsoo Kim, Dongwoo Song, Yongsoo Shim, Hyungbo Sandberg, Henrik Johansson, Karl H. Seoul Natl Univ Sci & Technol Dept Elect & Informat Engn Seoul South Korea Western Digital Res Milpitas CA 95035 USA Seoul Natl Univ Dept Comp Sci & Engn Seoul South Korea Seoul Natl Univ Dept Elect & Comp Engn ASRI Seoul South Korea KTH Royal Inst Technol Sch Elect Engn & Comp Sci & Digital Futures Stockholm Sweden

Encrypted control has been introduced to protect controller data by encryption at the stage of computation and communication, by performing the computation directly on encrypted data. In this article, we first review and categorize recent relevant studies on encrypted control. Approaches based on homomorphic encryption, multi-party computation, and secret sharing are introduced, compared, and then discussed with respect to computational complexity, communication load, enabled operations, security, and research directions. We proceed to discuss a current challenge in the application of homomorphic encryption to dynamic systems, where arithmetic operations other than integer addition and multiplication are limited. We also introduce a learning with errors based homomorphic cryptosystem called "Gentry-Sahai-Waters"scheme and discuss its benefits that allow for recursive multiplication of encrypted dynamic systems, without use of computationally expensive bootstrapping techniques.

关键词： Encrypted control Homomorphic encryption Bootstrapping Multi-party computation Secret sharing Dynamic system over encrypted data learning with errors

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Toward single-server private information retrieval protocol via learning with errors

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JOURNAL OF INFORMATION SECURITY AND APPLICATIONS 2017年第Jun. Pt.2期34卷 280-284页

作者： Li, Zengpeng Ma, Chunguang Wang, Ding Du, Gang Harbin Engn Univ Coll Comp Sci & Technol Harbin 150001 Heilongjiang Peoples R China Fujian Normal Univ Fujian Prov Key Lab Network Secur & Cryptol Fuzhou 350007 Fujian Peoples R China Peking Univ Sch Elect Engn & Comp Sci Beijing 100871 Peoples R China

At FOCS2011 Brakerski and Vaikuntanathan proposed a single-server LWE-based private information retrieval (abbreviated as PIR) protocol with a security reduction to hard standard lattice problems and nearly optimal communication complexity. However, Brakerski just described a generic PIR protocol that utilized a somewhat homomorphic encryption and an arbitrary symmetric encryption as building blocks, he did not instantiate the generic construction. In this work, we first modify Brakerski's construction without the evaluating key and construct a new PIR model. Moreover, we instantiate our new model via matrix FHE first proposed by Ryo et al. at PKC2015 and vector symmetric encryption scheme proposed in this work as building block. Then we optimize the Response operations and several other aspects of the scheme. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Private information retrieval Homomorphic encryption Multi-bit symmetric encryption learning with errors

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A New Approach for Non-Interactive Zero-Knowledge from learning with errors 2024

A New Approach for Non-Interactive Zero-Knowledge from Learn...

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56th Annual ACM Symposium on Theory of Computing (STOC)

作者： Waters, Brent Univ Texas Austin Austin TX 78712 USA NTT Res Sunnyvale CA 94085 USA

ISBN: (纸本)9798400703836

We put forward a new approach for achieving non-interactive zero-knowledge proofs (NIKZs) from the learning with errors (LWE) assumption (with subexponential modulus to noise ratio). We provide a LWE-based construction of a hidden bits generator that gives rise to a NIZK via the celebrated hidden bits paradigm. A notable feature of our construction is its simplicity. Our construction employs lattice trapdoors, but beyond that uses only simple operations. Unlike prior solutions, we do not rely on a correlation intractability argument nor do we utilize fully homomorphic encryption techniques. Our solution provides a new methodology that adds to the diversity of techniques for solving this fundamental problem.

关键词： Non-Interactive Zero Knowledge learning with errors

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A Secure Remote Password Protocol From The learning with errors Problem 21

A Secure Remote Password Protocol From The Learning With Err...

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21st IEEE International Conference on Trust, Security and Privacy in Computing and Communications (IEEE TrustCom)

作者： Li, Huapeng Wang, Baocheng North China Univ Technol Sch Informat Beijing Peoples R China

ISBN: (纸本)9781665494250

Secure Remote Password (SRP) protocol is an essential password-authenticated key exchange (PAKE) protocol based on the discrete logarithm problem (DLP). The protocol is specifically designed to obtain a session key and it has been widely used in various scenarios due to its attractive security features. In the SRP protocol, the server is not required to save any data directly associated with passwords. And this makes attackers who manage to corrupt the server fail to impersonate the client unless performing a brute-force search for the password. However, the development of quantum computing has potentially made classic DLP-based public-key cryptography schemes not secure, including the SRP protocol. So it is significant to design a quantum-resistant SRP protocol. In this paper, based on the original scheme, we propose a post-quantum SRP protocol from the learning with errors (LWE) problem. And we give rigorous proof and analyses on the correctness and security of the scheme. Besides being resistant to known quantum attacks, it maintains the various secure qualities of the original protocol.

关键词： Secure Remote Password Protocol Password-Authenticated Key Exchange learning with errors

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Identity-Based Lossy Encryption from learning with errors 10th

Identity-Based Lossy Encryption from Learning with Errors

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10th International Workshop on Security (IWSEC)

作者： He, Jingnan Li, Bao Lu, Xianhui Jia, Dingding Xue, Haiyang Sun, Xiaochao Chinese Acad Sci Inst Informat Engn State Key Lab Informat Secur Beijing Peoples R China Chinese Acad Sci Data Assurance & Commun Secur Res Ctr Beijing Peoples R China Univ Chinese Acad Sci Beijing Peoples R China

ISBN: (纸本)9783319224251;9783319224244

We extend the notion of lossy encryption to the scenario of identity-based encryption (IBE), and propose a new primitive called identity-based lossy encryption (IBLE). Similar as the case of lossy encryption, we show that IBLE can also achieve selective opening security. Finally, we present a construction of IBLE from the assumption of learning with errors.

关键词： Lossy encryption learning with errors Identity-based lossy encryption

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