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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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Non-commutative Ring learning with errors from Cyclic Algebras

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JOURNAL OF CRYPTOLOGY 2022年第3期35卷 1-67页

作者： Grover, Charles Mendelsohn, Andrew Ling, Cong Vehkalahti, Roope Imperial Coll London London England Univ Jyvaskyla Dept Math & Stat Jyvaskyla 40014 Finland

The learning with errors (LWE) problem is the fundamental backbone of modern lattice-based cryptography, allowing one to establish cryptography on the hardness of well-studied computational problems. However, schemes based on LWE are often impractical, so Ring LWE was introduced as a form of 'structured' LWE, trading off a hard to quantify loss of security for an increase in efficiency by working over a well-chosen ring. Another popular variant, Module LWE, generalizes this exchange by implementing a module structure over a ring. In this work, we introduce a novel variant of LWE over cyclic algebras (CLWE) to replicate the addition of the ring structure taking LWE to Ring LWE by adding cyclic structure to Module LWE. We show that the security reductions expected for an LWE problem hold, namely a reduction from certain structured lattice problems to the hardness of the decision variant of the CLWE problem (under the condition of constant rank d). As a contribution of theoretic interest, we view CLWE as the first variant of Ring LWE which supports non-commutative multiplication operations. This ring structure compares favorably with Module LWE, and naturally allows a larger message space for error correction coding.

关键词： Algebraic number theory Lattices learning with errors Non-commutative algebra Post-quantum cryptography

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

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JOURNAL OF INFORMATION SECURITY AND APPLICATIONS 2017年 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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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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Post Quantum learning with errors Problem Based Key Encapsulation Protocols and Matrix Vector Product 4

Post Quantum Learning With Errors Problem Based Key Encapsul...

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4th International Conference on Computer Science and Engineering (UBMK)

作者： Alkim, Erdem Yazar, Bilge Kagan Ondokuz Mayis Univ Bilgisayar Muhendisligi Samsun Turkey

ISBN: (纸本)9781728139647

For post quantum cryptography lattice based systems are among the most widely used and safest systems. One of the difficult problems on lattice based systems are based learning with errors (LWE) problem. LWE based protocols, because of the definition of the LWE problem, it contains matrix vector product. These multiplications in the protocols take a lot of time. Because, size of matrices used are very large. Therefore, these multiplications need to be accelerated, optimized and made with an acceptable time. In this study, FrodoKEM, Lizard, Emblem, Lotus key encapsulation mechanishms using LWE problem and matrix vector product in FrodoKEM are focused. A efficient matrix vector product library was created based on the productions in the FrodoKEM protocol and applied to other protocols and the results were examined.

关键词： Post Quantum Cryptography learning with errors Key Encapsulation Mechanishms Matrix Vector Product

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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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Analysis of Error Terms of Signatures Based on learning with errors 19th

Analysis of Error Terms of Signatures Based on Learning with...

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19th International Conference on Information Security and Cryptology (ICISC)

作者： Kim, Jeongsu Park, Suyong Kim, Seonggeun Jang, Busik Hahn, Sang Geun Jung, Sangim Roh, Dongyoung Korea Adv Inst Sci & Technol Dept Math Sci Daejeon South Korea Natl Secur Res Inst Daejeon South Korea

ISBN: (纸本)9783319531762;9783319531779

Lyubashevsky proposed a lattice-based digital signature scheme based on short integer solution (SIS) problem without using trapdoor matrices [12]. Bai and Galbraith showed that the hard problem in Lyubashevsky's scheme can be changed from SIS to SIS and learning with errors (LWE) [4]. Using this change, they could compress the signatures. But Bai and Galbraith's scheme had some additional rejection processes on its algorithms. These rejection processes decreased the acceptance rate of the signing algorithm. We showed mathematically that the rejection process in key generation algorithm of [4] is not necessary. Using this fact, we suggested a scheme modified from [4]'s scheme, and doubled the acceptance rate of the signing algorithm. Furthermore, our implementation results show that our scheme is two times faster than that of [4] on similar parameter settings.

关键词： Lattice Digital signature learning with errors LWE Discrete random variable Discrete Gaussian distribution

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