Lattice Reduction with Approximate Enumeration Oracles

Albrecht, M.; Bai, Shi; Li, Jianwei; Rowell, Joe

doi:10.1007/978-3-030-84245-1_25

Cited by 17 publications

(5 citation statements)

References 44 publications

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“…This approach has been experimentally observed to work in [33]. In the classical setting, the use of tightly controlled approximate SVP oracles was shown to provide an exponential speedup over enumeration with exact SVP oracles [34], but here we likely do not get to choose the looseness of our oracles. In principle, if we could characterise the distribution of the output of the algorithm and quantify the expected size of the vector, we could estimate the number of times the algorithm needs to be re-run before obtaining a shortest vector with good probability.…”

Section: Handling Approximate Solutionsmentioning

confidence: 99%

Variational quantum solutions to the Shortest Vector Problem

Albrecht¹,

Prokop

Shen

et al. 2023

Quantum

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A fundamental computational problem is to find a shortest non-zero vector in Euclidean lattices, a problem known as the Shortest Vector Problem (SVP). This problem is believed to be hard even on quantum computers and thus plays a pivotal role in post-quantum cryptography. In this work we explore how (efficiently) Noisy Intermediate Scale Quantum (NISQ) devices may be used to solve SVP. Specifically, we map the problem to that of finding the ground state of a suitable Hamiltonian. In particular, (i) we establish new bounds for lattice enumeration, this allows us to obtain new bounds (resp. estimates) for the number of qubits required per dimension for any lattices (resp. random q-ary lattices) to solve SVP; (ii) we exclude the zero vector from the optimization space by proposing (a) a different classical optimisation loop or alternatively (b) a new mapping to the Hamiltonian. These improvements allow us to solve SVP in dimension up to 28 in a quantum emulation, significantly more than what was previously achieved, even for special cases. Finally, we extrapolate the size of NISQ devices that is required to be able to solve instances of lattices that are hard even for the best classical algorithms and find that with approximately 103 noisy qubits such instances can be tackled.

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Section: Handling Approximate Solutionsmentioning

confidence: 99%

Variational quantum solutions to the Shortest Vector Problem

Albrecht¹,

Prokop

Shen

et al. 2023

Quantum

Self Cite

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“…There are two types of lattice reduction algorithms that may be used to implement the SVP "oracle," enumeration and sieving. Enumeration algorithms (see, for example, [298][299][300][301][302]) require small amounts of memory but have run times that are super-exponential in β . Sieving algorithms (see, for example [303][304][305]) have run times that are exponential in β , but also require an exponential amount of memory.…”

Section: On the Concrete Intractability Of Finding Short Lattice Vectorsmentioning

confidence: 99%

Status report on the third round of the NIST Post-Quantum Cryptography Standardization process

Moody¹

2022

110

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The National Institute of Standards and Technology is in the process of selecting public-key cryptographic algorithms through a public, competition-like process. The new public-key cryptography standards will specify additional digital signature, public-key encryption, and key-establishment algorithms to augment Federal Information Processing Standard (FIPS) 186-4, Digital Signature Standard (DSS), as well as NIST Special Publication (SP) 800-56A Revision 3, Recommendation for Pair-Wise Key-Establishment Schemes Using Discrete Logarithm Cryptography, and SP 800-56B Revision 2, Recommendation for Pair-Wise Key Establishment Using Integer Factorization Cryptography. It is intended that these algorithms will be capable of protecting sensitive information well into the foreseeable future, including after the advent of quantum computers. The first round of the NIST Post-Quantum Cryptography Standardization Process began in December 2017 with 69 candidate algorithms that met both the minimum acceptance criteria and submission requirements. The first round lasted until January 2019, during which candidate algorithms were evaluated based on their security, performance, and other characteristics. NIST selected 26 algorithms to advance to the second round for more analysis. The second round continued until July 2020, after which seven 'finalist' and eight 'alternate' candidate algorithms were selected to move into the third round. This report describes the evaluation and selection process, based on public feedback and internal review, of the third-round candidates. The report summarizes each of the 15 third-round candidate algorithms and identifies those selected for standardization, as well as those that will continue to be evaluated in a fourth round of analysis. The public-key encryption and key-establishment algorithm that will be standardized is CRYSTALS-Kyber. The digital signatures that will be standardized are CRYSTALS-Dilithium, Falcon, and SPHINCS+. While there are multiple signature algorithms selected, NIST recommends CRYSTALS-Dilithium as the primary algorithm to be implemented. In addition, four of the alternate key-establishment candidate algorithms will advance to a fourth round of evaluation: BIKE, Classic McEliece, HQC, and SIKE. These candidates are still being considered for future standardization. NIST will also issue a new Call for Proposals for public-key digital signature algorithms to augment and diversify its signature portfolio.

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“…This is a block-wise iterative algorithm for basis reduction that can be seen as a generalization of the LLL algorithm introduced in 1982 by Lenstra, Lenstra and Lovász [158]. Recently, several variants of the BKZ algorithm have been proposed [159]- [163]. In these algorithms, the time complexity and the outcome quality (i.e.…”

Section: A Lattice Reduction Algorithmsmentioning

confidence: 99%

Survey on Fully Homomorphic Encryption, Theory, and Applications

Marcolla

Sucasas

Manzano³

et al. 2022

Proc. IEEE

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Data privacy concerns are increasing significantly in the context of Internet of Things, cloud services, edge computing, artificial intelligence applications, and other applications enabled by next generation networks. Homomorphic Encryption addresses privacy challenges by enabling multiple operations to be performed on encrypted messages without decryption. This paper comprehensively addresses homomorphic encryption from both theoretical and practical perspectives. The paper delves into the mathematical foundations required to understand fully homomorphic encryption (FHE). It consequently covers design fundamentals and security properties of FHE, and describes the main FHE schemes based on various mathematical problems. On a more practical level, the paper presents a view on privacypreserving Machine Learning using homomorphic encryption, then surveys FHE at length from an engineering angle, covering the potential application of FHE in fog computing, and cloud computing services. It also provides a comprehensive analysis of existing state-of-the-art FHE libraries and tools, implemented in software and hardware, and the performance thereof.

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Lattice Reduction with Approximate Enumeration Oracles

Cited by 17 publications

References 44 publications

Variational quantum solutions to the Shortest Vector Problem

Variational quantum solutions to the Shortest Vector Problem

Status report on the third round of the NIST Post-Quantum Cryptography Standardization process

Survey on Fully Homomorphic Encryption, Theory, and Applications

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