Log-$$\mathcal {S}$$-unit Lattices Using Explicit Stickelberger Generators to Solve Approx Ideal-SVP

Bernard, Olivier; Lesavourey, Andrea; Nguyen, Tuong-Huy; Roux-Langlois, Adeline

doi:10.1007/978-3-031-22969-5_23

Cited by 7 publications

(1 citation statement)

References 37 publications

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“…One area of recent interest is S-unit attacks [324,325] and the "Twisted-PHS" algorithm [326][327][328], which can be viewed as generalizations of the quantum algorithm for γapproximate ideal SVP. These algorithms use a computationally-intensive pre-processing step to improve the quality of the solution that is found.…”

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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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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Some Easy Instances of Ideal-SVP and Implications on the Partial Vandermonde Knapsack Problem

Boudgoust

Gachon

Pellet-Mary

2022

Lecture Notes in Computer Science

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In this article, we generalize the works of Pan et al. (Eurocrypt'21) and Porter et al. (ArXiv'21) and provide a simple condition under which an ideal lattice defines an easy instance of the shortest vector problem. Namely, we show that the more automorphisms stabilize the ideal, the easier it is to find a short vector in it. This observation was already made for prime ideals in Galois fields, and we generalize it to any ideal (whose prime factors are not ramified) of any number field. We then provide a cryptographic application of this result by showing that particular instances of the partial Vandermonde knapsack problem, also known as partial Fourier recovery problem, can be solved classically in polynomial time. As a proof of concept, we implemented our attack and managed to solve those particular instances for concrete parameter settings proposed in the literature. For random instances, we can halve the lattice dimension with non-negligible probability.

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Log-$$\mathcal {S}$$-unit Lattices Using Explicit Stickelberger Generators to Solve Approx Ideal-SVP

Bernard¹,

Lesavourey²,

Nguyen³

et al. 2022

Lecture Notes in Computer Science

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In 2020, Bernard and Roux-Langlois introduced the Twisted-PHS algorithm to solve Approx-Svp for ideal lattices on any number field, based on the PHS algorithm by Pellet-Mary, Hanrot and Stehlé. They performed experiments for prime conductors cyclotomic fields of degrees at most 70, one of the main bottlenecks being the computation of a log-S-unit lattice which requires subexponential time. Our main contribution is to extend these experiments to cyclotomic fields of degree up to 210 for most conductors m. Building upon new results from Bernard and Kučera on the Stickelberger ideal, we use explicit generators to construct full-rank log-S-unit sublattices fulfilling the role of approximating the full Twisted-PHS lattice. In our best approximate regime, our results show that the Twisted-PHS algorithm outperforms, over our experimental range, the CDW algorithm by Cramer, Ducas and Wesolowski, and sometimes beats its asymptotic volumetric lower bound. Additionally, we use these explicit Stickelberger generators to remove almost all quantum steps in the CDW algorithm, under the mild restriction that the plus part of the class number verifies h + m ≤ O( √ m).

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Log-$$\mathcal {S}$$-unit Lattices Using Explicit Stickelberger Generators to Solve Approx Ideal-SVP

Cited by 7 publications

References 37 publications

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

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

Some Easy Instances of Ideal-SVP and Implications on the Partial Vandermonde Knapsack Problem

Log-$$\mathcal {S}$$-unit Lattices Using Explicit Stickelberger Generators to Solve Approx Ideal-SVP

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