MATHEMATICAL ANALYSIS OF QC-MDPC STRUCTURES IN BIKE V5.2 POST-QUANTUM KEY ENCAPSULATION SCHEME

  • Rosa Rosa Department of Cryptographic Engineering, Politeknik Siber dan Sandi Negara, Indonesia https://orcid.org/0009-0003-1654-2525
  • Sa'aadah Sajjana Carita Department of Cryptographic Engineering, Politeknik Siber dan Sandi Negara, Indonesia https://orcid.org/0000-0002-5198-085X
  • Nadia Paramita Retno Adiati Department of Cryptographic Engineering, Politeknik Siber dan Sandi Negara, Indonesia
  • Sri Rosdiana Department of Cryptographic Engineering, Politeknik Siber dan Sandi Negara, Indonesia https://orcid.org/0009-0004-6126-3004
DOI: https://doi.org/10.30598/barekengvol20iss2pp1061–1076
Keywords: BIKE v5.2, QC-MDPC, Key encapsulation, Mechanism, Post-quantum cryptography

Abstract

The security of the BIKE scheme depends on a complex mathematical structure built upon QC-MDPC codes. This scheme is constructed using the Niederreiter framework and the application of  transformation. Its security is based on the complexity of two main mathematical problems: the QCSD Problem and the QCCF Problem. The BIKE v5.2 scheme is the latest version of this scheme. This study aims to mathematically analyze the characteristics forming the BIKE v5.2, focusing on QC-MDPC codes, the Niederreiter framework, and the  transformation, as well as the QCSD and QCCF problems. The method used in this study is a systematic literature review combined with theoretical analysis. The study highlights how the interplay of these three components forms a rational and resilient design. Although the BIKE v5.2 scheme was not selected for standardization by NIST, it is still capable of producing an efficient, secure, and relevant KEM for post-quantum cryptography. Through mathematical analysis of the QC-MDPC construction, the formulation of the complex computational problems QCCF and QCSD, and the rational design of the Niederreiter framework with the  transformation, this study demonstrates that BIKE has a strong security foundation and resistance to both classical and quantum attacks.

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References

A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone, HANDBOOK OF APPLIED CRYPTOGRAPHY. CRC Press, 2018.doi: https://doi.org/10.1201/9780429466335

D. M. Pradana, “KAJIAN MATEMATIS MEKANISME ENKAPSULASI KUNCI KYBER.CCAKEM BERBASIS MASALAH LEARNING WITH ERROR,” 2023.

G. M. Raimondo and L. E. Locascio, MODULE-LATTICE-BASED KEY-ENCAPSULATION MECHANISM STANDARD. Gaithersburg,: National Institute of Standards and Technology, 2024. doi: https://doi.org/10.6028/NIST.FIPS.203.

N. Aragon., "BIKE: BIT FLIPPING KEY ENCAPSULATION," Internet Draft, 2024. [Online]. Available: https://bikesuite.org/docs/BIKE.pdf [Accessed: Oct. 22, 2024].

M. R. Nosouhi, S. W. A. Shah, L. Pan, and R. Doss, "BIT FLIPPING KEY ENCAPSULATION FOR THE POST-QUANTUM ERA," IEEE Access, vol. 11, pp. 56181–56195, 2023. doi: https://doi.org/10.1109/ACCESS.2023.3282928.

National Institute of Standards and Technology, "POST-QUANTUM CRYPTOGRAPHY PROJECT," 2024. [Online]. Available: https://csrc.nist.gov/Projects/post-quantum-cryptography/post-quantum-cryptography-standardization/Call-for-Proposals [Accessed: Oct. 26, 2024].

N. Aragon et al., "OFFICIAL WEB PAGE OF BIKE SUITE," 2024. [Online]. Available: https://bikesuite.org/ [Accessed: Dec. 26, 2024].

V. Vasseur, "POST-QUANTUM CRYPTOGRAPHY: A STUDY OF THE DECODING OF QC-MDPC CODES," Ph.D. dissertation, Université Paris Cité, Paris, France, 2021. [Online]. Available: https://theses.hal.science/tel-04523204v1

D. Hofheinz, K. Hövelmanns, and E. Kiltz, "A MODULAR ANALYSIS OF THE FUJISAKI-OKAMOTO TRANSFORMATION," in Proc. Theory of Cryptography Conf. (TCC), Cham, Switzerland: Springer, pp. 341–371, Nov. 2017. Doi: https://doi.org/10.1007/978-3-319-70500-2_12

S. W. A. Shah, M. R. Nosouhi, L. Pan, and R. Doss, "SoK: ON EFFICACY OF THE BGF DECODER FOR QC-MDPC-BASED QUANTUM-SAFE CRYPTOSYSTEMS," in Proc. 10th ACM Asia Public-Key Cryptography Workshop (AsiaPKC), Melbourne, Australia, Jul. 2023, pp. 2–9. doi: https://doi.org/10.1145/3591866.3593070.

G. Alagic et al., "STATUS REPORT ON THE FOURTH ROUND OF THE NIST POST-QUANTUM CRYPTOGRAPHY STANDARDIZATION PROCESS," NIST Internal Report (NIST IR) 8545, Gaithersburg, MD, USA, Mar. 2025. doi: https://doi.org/10.6028/NIST.IR.8545.

M. R. Nosouhi, .yed W. Shah, L. Pan, Y. Zolotavkin, A. Nanda, P. Gauravaram and R. Doss., "WEAK-KEY ANALYSIS FOR BIKE POST-QUANTUM KEY ENCAPSULATION MECHANISM," IEEE Trans. Inf. Forensics Secur., vol. 18, pp. 2160–2174, Apr. 2023. [Online]. Available: http://arxiv.org/abs/2204.13885

N. Drucker, S. Gueron, and D. Kostic, "BINDING BIKE ERRORS TO A KEY PAIR," in Proc. Cryptographers’ Track at RSA Conf. (CT-RSA), Cham, Switzerland: Springer, pp. 275–281, Jul. 2021.doi: https://doi.org/10.1007/978-3-030-78086-9_21

R. Misoczki, J.-P. Tillich, N. Sendrier, and P. S. L. M. Barreto, "MDPC-MCELIECE: NEW MCELIECE VARIANTS FROM MODERATE DENSITY PARITY-CHECK CODES," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Istanbul, Turkey, Jul. 2013, pp. 2069–2073.doi: https://doi.org/10.1109/ISIT.2013.6620590

M. Mahajan, B. Singh, A. Agrawal, and A. K. Mishral, "COMPARATIVE ANALYSIS OF BIT FLIPPING DECODERS IN BIKE PQC," in Lecture Notes in Networks and Systems, vol. 941. Cham, Switzerland: Springer, 2024, pp. 345–356. doi: https://doi.org/10.1007/978-981-99-9531-8_28.

T. W. Judson, ABSTRACT ALGEBRA: THEORY AND APPLICATIONS. Nacogdoches, TX: Stephen F. Austin State Univ., 2020. [Online]. Available: https://scholarworks.sfasu.edu/ebooks

W. C. Huffman and V. Pless, FUNDAMENTALS OF ERROR-CORRECTING CODES. Cambridge, U.K.: Cambridge Univ. Press, 2010.

I. Von Maurich, L. Heberle, and T. Güneysu, "IND-CCA SECURE HYBRID ENCRYPTION FROM QC-MDPC NIEDERREITER," in Proc. 7th Int. Workshop on Post-Quantum Cryptography (PQCrypto), Fukuoka, Japan, Feb. 2016. [Online]. Available: https://www.nsa.gov/ia/programs/suiteb doi: https://doi.org/10.1007/978-3-319-29360-8_1

T. Wang, A. Wang, and X. Wang, "EXPLORING DECRYPTION FAILURES OF BIKE: NEW CLASS OF WEAK KEYS AND KEY RECOVERY ATTACKS," in Proc. Int. Cryptology Conf. (CRYPTO), Santa Barbara, CA, USA, Aug. 2023. Doi: https://doi.org/10.1007/978-3-031-38548-3_3

Q. Guo, T. Johansson, and P. Stankovski, "A KEY RECOVERY ATTACK ON MDPC WITH CCA SECURITY USING DECODING ERRORS," in Advances in Cryptology – ASIACRYPT, Hanoi, Vietnam, Dec. 2016, pp. 789–815.doi: https://doi.org/10.1007/978-3-662-53887-6_29

M. J. Dworkin, "SHA-3 STANDARD: PERMUTATION-BASED HASH AND EXTENDABLE-OUTPUT FUNCTIONS," FEDERAL INFORMATION PROCESSING STANDARDS PUBLICATION (FIPS) 202, NIST, Gaithersburg, MD, USA, 2015. doi: https://doi.org/10.6028/NIST.FIPS.202.

Published
2026-01-26
How to Cite
[1]
R. Rosa, S. S. Carita, N. P. R. Adiati, and S. Rosdiana, “MATHEMATICAL ANALYSIS OF QC-MDPC STRUCTURES IN BIKE V5.2 POST-QUANTUM KEY ENCAPSULATION SCHEME”, BAREKENG: J. Math. & App., vol. 20, no. 2, pp. 1061–1076, Jan. 2026.
Issue
Vol 20 No 2 (2026): BAREKENG: Journal of Mathematics and Its Application
Section
Articles

Copyright (c) 2026 Rosa Rosa, Sa'aadah Sajjana Carita, Nadia Paramita Retno Adiati, Sri Rosdiana

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