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Error Floor Prediction with Markov Models for QC-MDPC Codes
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There is evidence that HQC has a sufficiently low DFR, and recent work indicates that with minor modifications BIKE achieves the same.
“There is evidence that HQC has a sufficiently low DFR and recent work indicates that with minor modifications, BIKE achieves the same [26].”
From Status report on the fourth round of the NIST post-quantum cryptography standardization process (2025) · cites this paper as [26]
The Summary of the Main Results reports strong evidence that BIKE can meet the DFR requirement with a modest key-size increase, a higher minimum threshold, and filtering of atypical keys. The Motivation mentions HQC’s selection for standardization but does not address whether HQC has a sufficiently low DFR. (LLM, conf=0.88)
Experimental — accuracy not yet measuredThe model predicted that increasing BIKE security level 1 block lengths from r = 12323 to r = 13477 would result in a conservative DFR estimate of 2^-129.5 for typical keys.
“Furthermore, the model predicted that increasing BIKE security level 1 block lengths from r = 12323 to r = 13477 would result in a conservative DFR estimate of 2 -129 . 5 for typical keys.”
From Status report on the fourth round of the NIST post-quantum cryptography standardization process (2025) · cites this paper as [26]
Section 1.3 reports that increasing the BIKE 1 block length from r = 12323 to r = 13477, with an improved decoding threshold, gives a predicted DFR below 2^-131.2 for a typical key. The cited copy does not report the claimed conservative estimate of 2^-129.5; the “Changing the Threshold Function” section gives approximately 2^-131.259 for r = 13477 instead. (LLM, conf=0.88)
Experimental — accuracy not yet measuredA model introduced by [26] was able to predict variations in DFR based on the structure of a key.
“Subsequently, a model introduced by [26] was able to predict variations in DFR based on the structure of a key.”
From Status report on the fourth round of the NIST post-quantum cryptography standardization process (2025) · cites this paper as [26]
In [26], a Markov model that tracked proximity to near codewords was used to predict the error floors for QC-MDPC codes under a generic iterative decoder.
“In [26], a Markov model that tracked proximity to near codewords was used to predict the error fl oors for QC-MDPC codes under a generic iterative decoder.”
From Status report on the fourth round of the NIST post-quantum cryptography standardization process (2025) · cites this paper as [26]
Results indicated that the error floor behavior in the BIKE DFR curves was dominated by convergence to these near codewords during failed decoding instances.
“Results indicated that the error fl oor behavior in the BIKE DFR curves was dominated by convergence to these near codewords during failed decoding instances [26].”
From Status report on the fourth round of the NIST post-quantum cryptography standardization process (2025) · cites this paper as [26]
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References 15 · Cited by 5
Research works printed but not linked: 17
- Arpin, S., Billingsley, T.R., Hast, D.R., Lau, J.B., Perlner, R.A., Robinson, A.: A study of error floor behavior in QC-MDPC codes. In: Cheon, J.H., Johansson, T. (eds.) PQCrypto 2022, LNCS, vol. 13512, pp. 89–103. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-17234-2_5 · DOI
- Annechini, A., Barenghi, A., Pelosi, G.: Bit-flipping decoder failure rate estimation for $$(v, w)-$$regular codes. In: Proceedings of IEEE International Symposium on Information Theory - ISIT, pp. 3374–3379. IEEE (2024) · DOI
- Annechini, A., Barenghi, A., Pelosi, G.: Bit-flipping decoder failure rate estimation for $$(v, w)-$$regular codes. CoRR, abs/2401.16919 (2024) · DOI
- Baldi, M., Barenghi, A., Chiaraluce, F., Pelosi, G., Santini, P.: Performance bounds for QC-MDPC codes decoders. In: Wachter-Zeh, A., Bartz, H., Liva, G. (eds.) CBCrypto 2021. LNCS, vol. 13150, pp. 95–122. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-98365-9_6 · DOI
- Drucker, N., Gueron, S., Kostic, D.: QC-MDPC decoders with several shades of gray. In: Ding, J., Tillich, J.-P. (eds.) PQCrypto 2020. LNCS, vol. 12100, pp. 35–50. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-44223-1_3 · DOI
- Gallager, R.G.: Low Density Parity Check Codes. M.I.T. Press, Cambridge, Massachusetts (1963) · DOI
- Guo, Q., Johansson, T., Stankovski, P.: A key recovery attack on MDPC with CCA security using decoding errors. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 789–815. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_29 · DOI
- Hashemi, Y., Banihashemi, A.H.: Characterization of elementary trapping sets in irregular LDPC codes and the corresponding efficient exhaustive search algorithms. IEEE Trans. Inf. Theory 64(5), 3411–3430 (2018) · DOI
- Hofheinz, D., Hövelmanns, K., Kiltz, E.: A modular analysis of the Fujisaki-Okamoto transformation. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 341–371. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_12 · DOI
- Nosouhi, M.R., et al.: Weak-key analysis for BIKE post-quantum key encapsulation mechanism. IEEE Trans. Inf. Forensics Secur. 18, 2160–2174 (2023) · DOI
- Richardson, T.: Error floors of LDPC codes. In: Proceedings of the 41th Annual Allerton Conference on Communication, Control, and Computing, vol. 41, pp. 1426–1435 (2003)
- Sendrier, N.: BIKE decoding failure – weak keys & error floors. In: Slides of the Panel Discussion of the Fifth PQC Standardization Conference, April 2024
- Sendrier, N., Vasseur, V.: On the decoding failure rate of QC-MDPC bit-flipping decoders. In: Ding, J., Steinwandt, R. (eds.) PQCrypto 2019. LNCS, vol. 11505, pp. 404–416. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-25510-7_22 · DOI
- Tanner, R.M.: A recursive approach to low complexity codes. IEEE Trans. Inform. Theory 27(5), 533–547 (1981) · DOI
- Vasseur, V.: Post-quantum cryptography: a study of the decoding of QC-MDPC codes. Ph.D. thesis, Université de Paris, March 2021
- Vasić, B., Chilappagari, S.K., Nguyen, D.V.: Chapter 6 - failures and error floors of iterative decoders. In: Declerq, D., Fossorier, M., Biglieri, E. (eds.) Academic Press Library in Mobile and Wireless Communications, pp. 299–341. Academic Press, Oxford (2014) · DOI
- Wang, T., Wang, A., Wang, X.: Exploring decryption failures of BIKE: new class of weak keys and key recovery attacks. In: Handschuh, H., Lysyanskaya, A. (eds.) CRYPTO 2023. LNCS, vol. 14083, pp. 70–100. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-38548-3_3 · DOI
Metadata
- Authors
- Arpin, S., Lau, J. B., Mesnard, A., Perlner, R., Robinson, A., Tillich, J.-P., & Vasseur, V.
- Affiliations
- Virginia Tech (US)
- KU Leuven (BE)
- Institut national de recherche en sciences et technologies du numérique (FR)
- National Institute of Standards and Technology (US)
- Thales (France) (FR)
- Corresponding author
- Arpin, Sarah
- Venue
- Lecture Notes in Computer Science
- Publisher
- Springer Nature Switzerland
- Published
- 2025
- Volume
- LNCS-16000
- Issue
- Part I
- Pages
- 221-252
- ISSN
- 0302-9743 (ISSN-L)
- 1611-3349
- OpenAlex
- W4413290327
- Cited by
- 5
- References
- 15
- Citations by year
- 2026: 4 · 2025: 1
- FWCI
- 4.74
- Type
- conference paper
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Lecture notes in computer science (publishedVersion)
- Lirias (submittedVersion, other-oa)
- HAL (submittedVersion, cc-by)
- Funders
- Agence Nationale de la Recherche (ANR-22-PETQ-0008)
- Topics
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Coding theory and cryptography
- Physical Sciences › Computer Science › Computer Networks and Communications › Cryptography and Data Security
- Physical Sciences › Computer Science › Information Systems › Cryptography and Residue Arithmetic
- Keywords
- QC-MDPC codes, error floor, Markov chain model, IND-CCA2 security, post-quantum cryptography, lattice-based encryption, key size
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