Skip to main content
SpringerLink
Log in

Efficiently Masking Polynomial Inversion at Arbitrary Order

  • Conference paper
  • First Online:
Post-Quantum Cryptography (PQCrypto 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13512))

Included in the following conference series:

Abstract

Physical side-channel analysis poses a huge threat to post-quantum cryptographic schemes implemented on embedded devices. Still, secure implementations are missing for many schemes. In this paper, we present an efficient solution for masked polynomial inversion, a main component of the key generation of multiple post-quantum Key Encapsulation Mechanisms (KEMs). For this, we introduce a polynomial-multiplicative masking scheme with efficient arbitrary order conversions from and to additive masking. Furthermore, we show how to integrate polynomial inversion and multiplication into the masking schemes to reduce costs considerably. We demonstrate the performance of our algorithms for two different post-quantum cryptographic schemes on the Cortex-M4. For NTRU, we measure an overhead of 35% for the first-order masked inversion compared to the unmasked inversion while for BIKE the overhead is as little as 11%. Lastly, we verify the security of our algorithms for the first masking order by measuring and performing a TVLA based side-channel analysis.

M. Krausz, G. Land and J. Brockmann—These authors contributed equally to this work.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Akkar, M.-L., Giraud, C.: An implementation of DES and AES, secure against some attacks. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 309–318. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44709-1_26

    Chapter  Google Scholar 

  2. Alagic, G., et al.: Status report on the first round of the NIST post-quantum cryptography standardization process. US Department of Commerce, National Institute of Standards and Technology (2019). https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=927303

  3. Aragon, N., et al.: BIKE: bit flipping key encapsulation (2021). https://bikesuite.org/files/v4.2/BIKE_Spec. 2021.07.26.1.pdf

  4. Van Beirendonck, M., D’anvers, J.-P., Karmakar, A., Balasch, J., Verbauwhede, I.: A Side-channel Resistant Implementation of SABER. ACM J. Emerg. Technol. Comput. Syst. (JETC) 17(2), 1–26 (2021)

    Article  Google Scholar 

  5. Bernstein, D.J., Chuengsatiansup, C., Lange, T., van Vredendaal, C.: NTRU prime: reducing attack surface at low cost. In: Adams, C., Camenisch, J. (eds.) SAC 2017. LNCS, vol. 10719, pp. 235–260. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72565-9_12

    Chapter  Google Scholar 

  6. Bernstein, D.J., Chuengsatiansup, C., Lange, T., van Vredendaal, C.: Ntru prime: round 3. Submission to the NIST PQC standardization process (2020). https://ntruprime.cr.yp.to

  7. Bernstein, D.J., Yang, B.-Y.: Fast constant-time GCD computation and modular inversion. IACR Trans. Cryptogr. Hardw. Embed. Syst. 2019(3), 340–398 (2019)

    Article  Google Scholar 

  8. Bos, J.W., Gourjon, M., Renes, J., Schneider, T., van Vredendaal, C.: Masking kyber: first- and higher-order implementations. IACR Trans. Cryptogr. Hardw. Embed. Syst. 2021(4), 173–214 (2021)

    Article  Google Scholar 

  9. Chari, S., Jutla, C.S., Rao, J.R., Rohatgi, P.: Towards sound approaches to counteract power-analysis attacks. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 398–412. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48405-1_26

    Chapter  Google Scholar 

  10. Chen, C., et al.: NTRU - algorithm specifications and supporting documentation. Brown University and Onboard security company, Wilmington USA (2019)

    Google Scholar 

  11. Chen, M.-S., Chou, T.: Classic McEliece on the arm cortex-M4. IACR Trans. Cryptogr. Hardw. Embed. Syst. 2021(3), 125–148 (2021)

    Article  Google Scholar 

  12. Chen, M.S., Güneysu, T., Krausz, M., Thoma, J.P.: Carry-less to BIKE faster. In: Ateniese, G., Venturi, D. (eds.) Applied Cryptography and Network Security - 20th International Conference, ACNS 2022, Rome, Italy, 20–23 June 2022, Proceedings, vol. 13269 of Lecture Notes in Computer Science, pp. 833–852. Springer, Heidelebrg (2022). https://doi.org/10.1007/978-3-031-09234-3_41

  13. Chung, C.M.M., Hwang, V., Kannwischer, M.J., Seiler, G., Shih, C.J., Yang, B.Y.: NTT multiplication for NTT-unfriendly rings new speed records for saber and NTRU on cortex-M4 and AVX2. IACR Trans. Cryptogr. Hardw. Embed. Syst. 2021(2), 159–188 (2021)

    Article  Google Scholar 

  14. Coron, J.S., Gérard, F., Montoya, S., Zeitoun, R.: High-order polynomial comparison and masking lattice-based encryption. Cryptology ePrint Archive (2021)

    Google Scholar 

  15. D’Anvers, J.P., Van Beirendonck, M., Verbauwhede, I.: Revisiting higher-order masked comparison for lattice-based cryptography: algorithms and bit-sliced implementations. IACR Cryptol. ePrint Arch., p. 110 (2022)

    Google Scholar 

  16. Ding, A.A., Zhang, L., Durvaux, F., Standaert, F.-X., Fei, Y.: Towards sound and optimal leakage detection procedure. In: Eisenbarth, T., Teglia, Y. (eds.) CARDIS 2017. LNCS, vol. 10728, pp. 105–122. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-75208-2_7

    Chapter  Google Scholar 

  17. Drucker, N., Gueron, S., Kostic, D.: Fast polynomial inversion for post quantum QC-MDPC cryptography. In: Dolev, S., Kolesnikov, V., Lodha, S., Weiss, G. (eds.) CSCML 2020. LNCS, vol. 12161, pp. 110–127. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-49785-9_8

    Chapter  Google Scholar 

  18. Fritzmann, T., et al.: Masked accelerators and instruction set extensions for post-quantum cryptography. IACR Trans. Cryptogr. Hardw. Embed. Syst. 2022(1), 414–460 (2021)

    Article  Google Scholar 

  19. Heinz, D., Kannwischer, M.J., Land, G., Pöppelmann, T., Schwabe, P., Sprenkels, D.: First-order masked kyber on ARM cortex-M4. Cryptology ePrint Archive, Report 2022/058 (2022). https://ia.cr/2022/058

  20. Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054868

    Chapter  Google Scholar 

  21. Hülsing, A., Rijneveld, J., Schanck, J., Schwabe, P.: High-speed key encapsulation from NTRU. In: Fischer, W., Homma, N. (eds.) CHES 2017. LNCS, vol. 10529, pp. 232–252. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66787-4_12

    Chapter  Google Scholar 

  22. Ishai, Y., Sahai, A., Wagner, D.: Private circuits: securing hardware against probing attacks. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 463–481. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_27

    Chapter  Google Scholar 

  23. Itoh, T., Tsujii, S.: A fast algorithm for computing multiplicative inverses in GF(2 \(\hat{\,}\)m) using normal bases. Inf. Comput. 78(3), 171–177 (1988)

    Article  MathSciNet  Google Scholar 

  24. Kannwischer, M.J., Rijneveld, J., Schwabe, P., Stoffelen, K.: PQM4: post-quantum crypto library for the ARM cortex-M4. https://github.com/mupq/pqm4

  25. Karabulut, E., Alkim, E., Aysu, A.: Single-trace side-channel attacks on \(\omega \)-small polynomial sampling: with applications to NTRU, NTRU prime, and crystals-dilithium. In: HOST, pp. 35–45. IEEE (2021)

    Google Scholar 

  26. Kundu, S., D’Anvers, J.P., Van Beirendonck, M., Karmakar, A., Verbauwhede, I.: Higher-order masked Saber. IACR Cryptol. ePrint Arch., 389 (2022)

    Google Scholar 

  27. Li, C.L.: Implementation of polynomial modular inversion in lattice based cryptography on ARM (2021)

    Google Scholar 

  28. Melchor, C.A., et al.: Hamming Quasi-Cyclic (HQC) - Third round version (2021)

    Google Scholar 

  29. Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986). https://doi.org/10.1007/3-540-39799-X_31

    Chapter  Google Scholar 

  30. Mujdei, C., et al.: Side-channel analysis of lattice-based post-quantum cryptography: exploiting polynomial multiplication. IACR Cryptol. ePrint Arch., 474 (2022)

    Google Scholar 

  31. Richter-Brockmann, J., Chen, M.-S., Ghosh, S., Güneysu, T.: Racing BIKE: improved polynomial multiplication and inversion in hardware. IACR Trans. Cryptogr. Hardw. Embed. Syst. 2022(1), 557–588 (2022)

    Google Scholar 

  32. Richter-Brockmann, J., Mono, J., Güneysu, T.: Folding BIKE: scalable hardware implementation for reconfigurable devices. IEEE Trans. Comput. 71(5), 1204–1215 (2022)

    Article  Google Scholar 

  33. Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)

    Article  MathSciNet  Google Scholar 

  34. Schneider, T., Moradi, A.: Leakage assessment methodology - a clear roadmap for side-channel evaluations. In: Güneysu, T., Handschuh, H. (eds.) CHES 2015. LNCS, vol. 9293, pp. 495–513. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48324-4_25

    Chapter  Google Scholar 

  35. Schneider, T., Paglialonga, C., Oder, T., Güneysu, T.: Efficiently masking binomial sampling at arbitrary orders for lattice-based crypto. In: Lin, D., Sako, K. (eds.) PKC 2019. LNCS, vol. 11443, pp. 534–564. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17259-6_18

    Chapter  Google Scholar 

  36. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)

    Article  MathSciNet  Google Scholar 

  37. Sim, B.Y., Kwon, J., Choi, K.Y., Cho, J., Park, A., Han, D.G.: Novel side-channel attacks on quasi-cyclic code-based cryptography. IACR Trans. Cryptogr. Hardw. Embed. Syst., 180–212 (2019)

    Google Scholar 

Download references

Acknowledgments

This work was supported by the German Research Foundation under Germany’s Excellence Strategy – EXC 2092 CASA – 390781972, through the H2020 project PROMETHEUS (grant agreement ID 780701), and by the Federal Ministry of Education and Research of Germany through the QuantumRISC (16KIS1038) and PQC4Med (16KIS1044) projects.

Author information

Authors and Affiliations

  1. Horst Görtz Institute for IT Security, Ruhr University Bochum, Bochum, Germany

    Markus Krausz, Georg Land, Jan Richter-Brockmann & Tim Güneysu

  2. DFKI GmbH, Cyber-Physical Systems, Bremen, Germany

    Georg Land & Tim Güneysu

Authors
  1. Markus Krausz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georg Land
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Richter-Brockmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tim Güneysu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georg Land .

Editor information

Editors and Affiliations

  1. Seoul National University, Seoul, Korea (Republic of)

    Jung Hee Cheon

  2. Lund University, Lund, Sweden

    Thomas Johansson

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Krausz, M., Land, G., Richter-Brockmann, J., Güneysu, T. (2022). Efficiently Masking Polynomial Inversion at Arbitrary Order. In: Cheon, J.H., Johansson, T. (eds) Post-Quantum Cryptography. PQCrypto 2022. Lecture Notes in Computer Science, vol 13512. Springer, Cham. https://doi.org/10.1007/978-3-031-17234-2_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-17234-2_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-17233-5

  • Online ISBN: 978-3-031-17234-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation