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On the dimension and structure of the square of the dual of a Goppa code
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During the fourth round, there has been significant progress in cryptanalysis techniques applicable to key recovery and to distinguishing a Goppa code from a random linear code.
“During the fourth round, there has been significant progress in cryptanalysis techniques that are applicable to key recovery and the related problem of distinguishing a Goppa code from a random linear code [46, 66-69].”
From Status report on the fourth round of the NIST post-quantum cryptography standardization process (2025) · cites this paper as [66]
The Introduction (pp. 1–2) and Conclusion (p. 16) document progress on distinguishing high-rate Goppa codes from random codes. But the Conclusion says turning this distinguisher into a Goppa-code key-recovery attack remains a hope, not an achieved result. The cited passages also do not place the progress specifically during the fourth round. (LLM, conf=0.87)
Experimental — accuracy not yet measuredAlgebraic attacks targeting the structure of Goppa codes that achieve either key recovery or a distinguisher from a random linear code have remained an active area of research.
“Nonetheless, algebraic attacks that target the structure of Goppa codes and achieve either key recovery or a distinguisher from a random linear code have remained an active area of research [46, 65-69].”
From Status report on the fourth round of the NIST post-quantum cryptography standardization process (2025) · cites this paper as [66]
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A method for finding codewords of small weight
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Status report on the fourth round of the NIST post-quantum cryptography standardization process
Alagic, G., Bros, M., Ciadoux, P., Cooper, D., Dang, Q., Dang, T., Kelsey, J., Lichtinger, J., Liu, Y.-K., Miller, C., et al. (2025) · doi:10.6028/nist.ir.8545 · Cited by 211
Metadata
- Authors
- Mora, R., & Tillich, J.-P.
- Affiliations
- Institut national de recherche en sciences et technologies du numérique (FR)
- Corresponding author
- Mora, Rocco
- Venue
- Designs, Codes and Cryptography
- Published
- 2022-11-22
- Volume
- 91
- Issue
- 4
- Pages
- 1351-1372
- ISSN
- 0925-1022 (ISSN-L)
- 1573-7586
- OpenAlex
- W3215246204
- Cited by
- 18
- References
- 1
- Citations by year
- 2026: 5 · 2025: 3 · 2024: 3 · 2023: 7
- FWCI
- 1.07
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Designs Codes and Cryptography (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (submittedVersion, other-oa)
- HAL (submittedVersion, cc-by)
- arXiv (Cornell University) (cc-by)
- Funders
- Agence Nationale de la Recherche (ANR-17-CE39-0007, ANR-21-CE39-0009)
- Topics
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Coding theory and cryptography
- Physical Sciences › Computer Science › Information Systems › Cryptography and Residue Arithmetic
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Polynomial and algebraic computation
- Keywords
- Goppa codes, alternant codes, dual codes, dimension bounds, TRACE code, non-binary codes
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