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A Note on the Implementation of the Number Theoretic Transform

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10655))

Abstract

The Number Theoretic Transform (NTT) is a time critical function required by many post-quantum cryptographic protocols based on lattices. For example it is commonly used in the context of the Ring Learning With Errors problem (RLWE), which is a popular basis for post-quantum key exchange, digital signature, and encryption. Here we apply a simple methodology to convert the NTT and its inverse from a mathematically correct (but side-channel vulnerable) description, to an efficient constant-time and side-channel resistant version.

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Notes

  1. 1.

    http://indigo.ie/~mscott/ntt_ref.c.

References

  1. Ring learning with errors parameters (2017). http://www.ringlwe.info/parameters-for-rlwe.html

  2. Alkim, E., Ducas, L., Poppelmann, T., Schwabe, P.: Post-quantum key exchange - a new hope. In: 25th Usenix Security Symposium, pp. 327–343 (2016)

    Google Scholar 

  3. Alkim, E., Jakubeit, P., Schwabe, P.: NewHope on ARM Cortex-M. In: Carlet, C., Hasan, M.A., Saraswat, V. (eds.) SPACE 2016. LNCS, vol. 10076, pp. 332–349. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-49445-6_19

    Chapter  Google Scholar 

  4. Barrett, P.: Implementing the Rivest Shamir and Adleman public key encryption algorithm on a standard digital signal processor. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 311–323. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-47721-7_24

    Google Scholar 

  5. Bernstein, D., Breitner, J., Genkin, D., Bruinderink, L.G., Heninger, N., Lange, T., van Vredendaal, C., Yarom, Y.: Sliding right into disaster: left-to-right sliding windows leak. Cryptology ePrint Archive, Report 2017/627 (2017). http://eprint.iacr.org/2017/627

  6. Fog, A.: Instruction tables: lists of instruction latencies, throughputs and microoperation breakdowns for intel, AMD and VIA CPUs (2017). http://www.agner.org/optimize/

  7. Güneysu, T., Oder, T., Pöppelmann, T., Schwabe, P.: Software speed records for lattice-based signatures. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 67–82. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38616-9_5

    Chapter  Google Scholar 

  8. Harvey, D.: Faster arithmetic for number-theoretic transforms. J. Symb. Comput. 60, 113–119 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Longa, P., Naehrig, M.: Speeding up the number theoretic transform for faster ideal lattice-based cryptography. In: Foresti, S., Persiano, G. (eds.) CANS 2016. LNCS, vol. 10052, pp. 124–139. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-48965-0_8

    Chapter  Google Scholar 

  10. Montgomery, P.: Modular multiplication without trial division. Math. Comput. 44(170), 519–521 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pöppelmann, T., Güneysu, T.: Towards efficient arithmetic for lattice-based cryptography on reconfigurable hardware. In: Hevia, A., Neven, G. (eds.) LATINCRYPT 2012. LNCS, vol. 7533, pp. 139–158. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33481-8_8

    Chapter  Google Scholar 

  12. Scott, M.: Slothful reduction. Cryptology ePrint Archive, Report 2017/437 (2017). http://eprint.iacr.org/2017/437

  13. Streit, S., De Santis, F.: Post-quantum key exchange on ARMv8-A - a new hope for NEON made simple. Cryptology ePrint Archive, Report 2017/388 (2017). http://eprint.iacr.org/2017/388

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Correspondence to Michael Scott .

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Scott, M. (2017). A Note on the Implementation of the Number Theoretic Transform. In: O'Neill, M. (eds) Cryptography and Coding. IMACC 2017. Lecture Notes in Computer Science(), vol 10655. Springer, Cham. https://doi.org/10.1007/978-3-319-71045-7_13

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  • DOI: https://doi.org/10.1007/978-3-319-71045-7_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71044-0

  • Online ISBN: 978-3-319-71045-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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