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Inner product encryption from ring learning with errors

Cybersecurity 2018年第1期1卷 922-932页

作者： Shisen Fang Shaojun Yang Yuexin Zhang The Fujian Provincial Key Laboratory of Network Security and Cryptology College of Mathematics and InformaticsFujian Normal University350108Fuzhou PRChina

The functional encryption scheme designed using the lattice can realize fine-grained encryption and it can resist quantum ***,the sizes of the keys and ciphertexts in cryptographic applications based on learning with errors are large,which makes the algorithm ***,we construct a functional encryption for inner product predicates scheme by improving the learning with errors scheme of Agrawal et al.[Asiacrypt 2011],and its security relies on the difficulty assumption of ring learning with *** construction can reduce the sizes of the keys and ciphertexts compared with the learning with errors scheme.

关键词： Functional encryption Inner product encryption Lattices ring learning with errors

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Fast multiplication and the PLWE-RLWE equivalence for an infinite family of maximal real subfields of cyclotomic fields

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DESIGNS CODES AND CRYPTOGRAPHY 2025年 1-23页

作者： Ahola, Joonas Blanco-Chacon, Ivan Bolanos, Wilmar Haavikko, Antti Hollanti, Camilla Sanchez-Ledesma, Rodrigo M. Huawei Technol Oy Helsinki Finland Univ Alcala Alcala De Henares Spain Aalto Univ Espoo Finland Univ Complutense Madrid Indra Madrid Spain Fdn Univ Konrad Lorenz Bogota Colombia

We prove the equivalence between the ring learning with errors (RLWE) and the Polynomial learning With errors (PLWE) problems for the maximal totally real subfield of the 2r3s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<^>r 3<^>s$$\end{document}th cyclotomic field for r >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \ge 3$$\end{document} and s >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \ge 1$$\end{document}. Moreover, we describe a fast algorithm for computing the product of two elements in the ring of integers of these subfields. This multiplication algorithm has quasilinear complexity in the dimension of the field, as it makes use of the fast Discrete Cosine Transform (DCT). Our approach assumes that the two input polynomials are given in a basis of Chebyshev-like polynomials, in contrast to the customary power basis. To validate this assumption, we prove that the change of basis from the power basis to the Chebyshev-like basis can be computed with O(nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(n \log n)$$\end{document} arithmetic operations, where n is the problem dimension. Finally, we provide a heuristic and theoretical comparison of the vulnerability to some attacks for the pth cyclotomic field versu

关键词： Number-Theoretic Transform Discrete Cosine Transform Fast Multiplication ring learning with errors Polynomial learning With errors Condition Number Abelian Number Fields

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Design and analysis of a post-quantum secure three party authenticated key agreement protocol based on ring learning with error for mobile device

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JOURNAL OF SUPERCOMPUTING 2025年第1期81卷 1-28页

作者： Kumar, Uddeshaya Garg, Manish Chaudhary, Dharminder LNM Inst Informat Technol Dept Math Jaipur 302031 Rajasthan India Amrita Vishwa Vidyapeetham Amrita Sch Comp Dept Comp Sci & Engn Chennai 601103 Tamil Nadu India

A three party authenticated key agreement protocol (3PAKA) enables two entities to establish a session key with the assistance of a dedicated server via an insecure channel. Recently, Islam et al. (J Inf Secur Appl 6363:103026, 2021) proposed a post-quantum secure 3PAKA scheme based on ring-LWE and proved their protocol could fend off several typical security threats. Later, Rewal et al. (J Inf Secur Appl 75:103505, 2023) pointed out that their scheme is susceptible to password guessing attacks. However, we have found that Islam et al. protocol is not only insecure against password guessing attacks but also lacks user privacy and is susceptible to impersonation attacks. We also find that Rewal et al. scheme does not provide anonymity to users and uses time stamps for key freshness, which might cause a clock synchronization problem. With that, there is a flaw in the password update phase of Rewal's scheme. We have proposed a 3PAKA protocol to overcome the vulnerabilities in the protocols mentioned above. We demonstrate that the new scheme strengthens security and mitigates all the existing defects of Islam et al. and Rawel et al. protocols. With this, the security comparison and performance analysis show that the proposed scheme offers a more effective solution than the existing schemes.

关键词： Authenticated key agreement Anonymity ring learning with errors Information security Cryptanalysis

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Fast polynomial arithmetic in homomorphic encryption with cyclo-multiquadratic fields

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2025年 1-35页

作者： Blanco-Chacon, Ivan Pedrouzo-Ulloa, Alberto Nchiwo, Rahinatou Y. Njah Barbero-Lucas, Beatriz Univ Alcala Dept Phys & Math Alcala De Henares Spain Univ Vigo atlanttic Vigo 36310 Galicia Spain Aalto Univ Dept Math & Syst Anal Espoo Finland Univ Coll Dublin Sch Math & Stat Dublin Ireland

We discuss the advantages and limitations of cyclotomic fields to have fast polynomial arithmetic within homomorphic encryption, and show how these limitations can be overcome by replacing cyclotomic fields by a family that we refer to as cyclo-multiquadratic. This family is of particular interest due to its arithmetic efficiency properties and to the fact that the Polynomial learning with errors (PLWE) and ring learning with errors (RLWE) problems are equivalent for it. Likewise, we provide exact expressions for the condition number for any cyclotomic field, but under what we call the twisted power basis. As a tool for our result, we obtain refined polynomial upper bounds for the condition number of cyclotomic fields with up to 6 different primes dividing the conductor. From a more practical side, we also show that for this family, swapping between NTT (Number Theoretic Transform) and coefficient representations can be achieved at least twice faster than for the usual cyclotomic family.

关键词： ring learning with errors Polynomial learning with errors Condition Number Cyclotomic polynomials Homomorphic Encryption Number Theoretic Transforms

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Efficient Zero-Knowledge Proofs for Commitments from learning with errors over rings 20th

Efficient Zero-Knowledge Proofs for Commitments from Learnin...

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20th European Symposium on Research in Computer Security (ESORICS)

作者： Benhamouda, Fabrice Krenn, Stephan Lyubashevsky, Vadim Pietrzak, Krzysztof ENS CNRS INRIA 46 Rue Ulm F-75230 Paris 05 France PSL Paris France AIT Austrian Inst Technol GmbH Vienna Austria ENS INRIA 46 Rue Ulm F-75230 Paris 05 France IST Austria Klosterneuburg Austria

ISBN: (纸本)9783319241746;9783319241739

We extend a commitment scheme based on the learning with errors over rings (RLWE) problem, and present efficient companion zero-knowledge proofs of knowledge. Our scheme maps elements from the ring (or equivalently, n elements from F-q) to a small constant number of ring elements. We then construct S-protocols for proving, in a zero-knowledge manner, knowledge of the message contained in a commitment. We are able to further extend our basic protocol to allow us to prove additive and multiplicative relations among committed values. Our protocols have a communication complexity of O(Mn log q) and achieve a negligible knowledge error in one run. Here M is the constant from a rejection sampling technique that we employ, and can be set close to 1 by adjusting other parameters. Previously known S-protocols for LWE-related languages only achieved a noticeable or even constant knowledge error (thus requiring many repetitions of the protocol), or relied on "smudging" out the error (which necessitates working over large fields, resulting in poor efficiency).

关键词： Commitment schemes ring learning with errors ZeroKnowledge Proofs of Knowledge

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Double Batch for RLWE-Based Leveled Fully Homomorphic Encryption

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Chinese Journal of Electronics 2015年第3期24卷 661-666页

作者： CHEN Hu HU Yupu LIAN Zhizhu State Key Laboratory of Integrated Service Networks Xidian University

To further improve the efficiency of Fully homomorphic encryption(FHE), a leveled FHE scheme based on the ring learning with errors(RLWE) problem is put forward by simultaneously applying both batch techniques available. Our scheme therefore allows double packing many plaintext values into each ciphertext to support single-instruction-multiple-data-type operations, which effectively reduces the ciphertext expansion ratio. An efficient evolutionary method for achieving arbitrary homomorphic permutation operations on a packed ciphertext is also provided by using several given key-switching ***, a few new operations are introduced, with which not only to describe the key switching process in our batch setting clearly, but also to analyze the noise growth conveniently.

关键词： Fully homomorphic encryption ring learning with errors Batch technique Circular security

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VLSI Design and FPGA Implementation of an NTT Hardware Accelerator for Homomorphic SEAL-Embedded Library

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IEEE ACCESS 2023年 11卷 72498-72508页

作者： Di Matteo, Stefano Lo Gerfo, Matteo Saponara, Sergio Univ Pisa Dept Informat Engn I-56126 Pisa Italy

Homomorphic Encryption (HE) allows performing specific algebraic computations on encrypted data without the need for decryption. For this reason, HE is emerging as a strong privacy-preserving solution in cloud computing environments since it allows to keep data secure even in the case the cloud server is not trusted. HE libraries such as Microsoft SEAL have been recently released;however, such libraries are not specifically designed for resource-constrained platforms and they are often expensive in terms of computational resources and memory consumption, which limits their usage in edge devices. This limitation is contained by the SEAL-Embedded library, the first C-based HE library specifically designed for embedded platforms. In this article, we propose a hardware accelerator specifically designed for the SEAL-Embedded library and its implementation of the CKKS scheme: the proposed hardware presents a configurable Number Theoretic Transform (NTT) unit for all the polynomial degrees available on the SEAL-Embedded, a memory architecture able to reduce the I/O latency and a dedicated module for the generation of roots of unity. A complete system that includes a 32-bit RISC-V (RI5CY) processor has been implemented on a Xilinx ZCU106 FPGA board to test the functionality of the hardware accelerator and to measure performance improvements. The results showed a speed-up of around x1000 with the hardware acceleration respect to the pure software implementation of the SEAL-Embedded library for the symmetric encryption function.

关键词： & nbsp Hardware accelerator homomorphic encryption number theoretic transform ring learning with errors RISC-V SEAL-Embedded FPGA

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Algebraic Aspects of Solving ring-LWE, Including ring-Based Improvements in the Blum-Kalai-Wasserman Algorithm

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SIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY 2021年第2期5卷 366-387页

作者： Stange, Katherine E. Univ Colorado Dept Math Boulder CO 80309 USA

We provide a reduction of the ring-LWE problem to ring-LWE problems in subrings, in the presence of samples of a restricted form (i.e., (a, b) such that a is restricted to a multiplicative coset of the subring). To create and exploit such restricted samples, we propose ring-BKW, a version of the Blum-Kalai-Wasserman (BKW) algorithm which respects the ring structure. Off-the-shelf BKW dimension reduction (including coded-BKW and sieving) can be used for the reduction phase. Its primary advantage is that there is no need for back-substitution, and the solving/hypothesis-testing phase can be parallelized. We also present a method to exploit symmetry to reduce table sizes, samples needed, and runtime during the reduction phase. The results apply to 2-power cyclotomic ring-LWE with parameters proposed for practical use (including all splitting types).

关键词： cryptography ring learning with errors Blum-Kalai-Wasserman lattice-based cryptography cyclotomic fields learning with errors

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APAS: Application-Specific Accelerators for RLWE-Based Homomorphic Linear Transformations

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IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY 2021年 16卷 4663-4678页

作者： Bian, Song Kundi, Dur E. Shahwar Hirozawa, Kazuma Liu, Weiqiang Sato, Takashi Kyoto Univ Dept Commun & Comp Engn Kyoto 6068501 Japan Nanjing Univ Aeronaut & Astronaut Coll Elect & Informat Engn Nanjing 211106 Peoples R China Natl Univ Sci & Technol Dept Elect Engn Islamabad 44000 Pakistan

Recently, the application of multi-party secure computing schemes based on homomorphic encryption in the field of machine learning attracts attentions across the research fields. Previous studies have demonstrated that secure protocols adopting packed additive homomorphic encryption (PAHE) schemes based on the ring learning with errors (RLWE) problem exhibit significant practical merits, and are particularly promising in enabling efficient secure inference in machine-learning-as-a-service applications. In this work, we introduce a new technique for performing homomorphic linear transformation (HLT) over PAHE ciphertexts. Using the proposed HLT technique, homomorphic convolutions and inner products can be executed without the use of number theoretic transform and the rotate-and-add algorithms that were proposed in existing works. To maximize the efficiency of the HLT technique, we propose APAS, a hardware-software co-design framework consisting of approximate arithmetic units for the hardware acceleration of HLT. In the experiments, we use actual neural network architectures as benchmarks to show that APAS can improve the computational and communicational efficiency of homomorphic convolution by 8x and 3x, respectively, with an energy reduction of up to 26x as compared to the ASIC implementations of existing methods.

关键词： Cryptography Protocols Computer architecture Transforms Standards Libraries Lattices Homomorphic encryption ring learning with errors secure inference neural networks homomorphic linear transformation number theoretic transform

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A New Constant-Size Group Signature Scheme From Lattices

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IEEE ACCESS 2020年 8卷 10198-10207页

作者： Luo, Qin Jiang, Chun-Yang Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Jilin Univ Math Changchun 130012 Peoples R China Jilin Univ Inst Math Changchun 130012 Peoples R China

A lattice-based group signature scheme (LGSS) is an active cryptographic primitive, where each group member can sign messages anonymously in the name of the entire group and each valid signature should be traced to some group member on the lattice. In each LGSS, the size of the group signature usually depends on the number of group members and the security parameter. Thus, designing a constant-size LGSS is an interesting problem. At PKC 2018, Ling, Nguyen, Wang and Xu presented the first constant-size group signature scheme under lattice assumptions. Its design is based on a zero-knowledge argument of the knowledge of a valid message-signature pair for the Ducas-Micciancio signature scheme, which follows the sign-then-encrypt-then-prove protocol. In contrast to this work, we construct a new constant-size LGSS. The scheme adopts the sign-hybrid-encrypt approach and makes use of the Lyubashevsky signature scheme. Our work is efficient in the signing algorithm, more precise on the open algorithm and shorter in public key, secret key and signature size than previous studies. Furthermore, we prove that the scheme has full anonymity and full traceability under the ring learning with errors and ring Short Integer Solution assumptions in the random oracle model.

关键词： Group signatures lattices Lyubashevsky signature scheme ring learning with errors ring short integer solution

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