Skip to main content
SpringerLink
Log in

Count-then-Permute: A Precision-Free Alternative to Inversion Sampling

  • Conference paper
  • First Online:
Topics in Cryptology – CT-RSA 2018 (CT-RSA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10808))

Included in the following conference series:

  • 1072 Accesses

Abstract

The sampling from a discrete probability distribution on computers is an old problem having a wide variety of applications. The inversion sampling which uses the cumulative probability table is quite popular method for discrete distribution sampling. One drawback of inversion sampling (and most of other generic methods) is that it’s table size and sampling time depends on the precision we require. This can be problematic, since the precision can be quite high, e.g., 256 bits or even more, in particular for cryptographic purpose. In this paper, we present a novel sampling method which we call counter-then-permute (CP) sampler. Our proposal has a unique feature that its time and memory for on-line sampling phase does not depend on the precision, and can be faster and smaller than inversion sampling, which was often the most efficient one, depending on the relationship between the precision and the number of samples we want. Our proposal uses a block cipher as an efficient, computationally-secure instantiation of uniform sampling without replacement, also known as a pseudorandom permutation (PRP) in the cryptographic terminology, and pre-processing based on a recent polynomial-time exact sampling for binomial distribution. We also show some experimental results of CP sampler for discrete Gaussian distributions, which are typically used by lattice-based cryptographic schemes.

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

Notes

  1. 1.

    Theoretically, a PRP can be built on any PRG [Gol99]. Thus, in principle we did not introduce any new computational assumption from PRG assumption.

  2. 2.

    Any other non-repeating sequence could be used as well.

  3. 3.

    I.e. queries are not restricted to counter \(1,2,\dots ,\) and may be adaptively chosen. Such queries are also called nonce.

  4. 4.

    More precisely, [BKP+14] showed that \(\mathcal{B}(N,1/2)\) can be exactly sampled expected O(1) time without pre-computation, and [FT15] showed that, using \(\mathcal{B}(N,1/2)\)-sampler as a black-box, one can sample from \(\mathcal{B}(N,p)\) for any p with expected \(\log N\) calls to the \(\mathcal{B}(N,1/2)\)-sampler. Farach-Colton and Tsai also showed O(1)-time sampling with high probability is possible with \(O((\log N)^\epsilon )\)-time preprocessing, for any \(\epsilon >0\).

References

  1. Banik, S., Bogdanov, A., Isobe, T., Shibutani, K., Hiwatari, H., Akishita, T., Regazzoni, F.: Midori: a block cipher for low energy. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9453, pp. 411–436. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48800-3_17

    Chapter  Google Scholar 

  2. Borghoff, J., et al.: PRINCE – a low-latency block cipher for pervasive computing applications. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 208–225. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-34961-4_14

    Chapter  Google Scholar 

  3. Buchmann, J., Cabarcas, D., Göpfert, F., Hülsing, A., Weiden, P.: Discrete Ziggurat: a time-memory trade-off for sampling from a Gaussian distribution over the integers. In: Lange, T., Lauter, K., Lisoněk, P. (eds.) SAC 2013. LNCS, vol. 8282, pp. 402–417. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43414-7_20

    Chapter  Google Scholar 

  4. Bellare, M., Desai, A., Jokipii, E., Rogaway, P.: A concrete security treatment of symmetric encryption. In: Symposium on Foundations of Computer Science - FOCS 1997, pp. 394–403. IEEE Computer Society (1997)

    Google Scholar 

  5. Bai, S., Galbraith, S.D.: An improved compression technique for signatures based on learning with errors. In: Benaloh, J. (ed.) CT-RSA 2014. LNCS, vol. 8366, pp. 28–47. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04852-9_2

    Chapter  Google Scholar 

  6. Bogdanov, A., Knudsen, L.R., Leander, G., Paar, C., Poschmann, A., Robshaw, M.J.B., Seurin, Y., Vikkelsoe, C.: PRESENT: an ultra-lightweight block cipher. In: Paillier, P., Verbauwhede, I. (eds.) CHES 2007. LNCS, vol. 4727, pp. 450–466. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74735-2_31

    Chapter  Google Scholar 

  7. Bringmann, K., Kuhn, F., Panagiotou, K., Peter, U., Thomas, H.: Internal DLA: efficient simulation of a physical growth model. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 247–258. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43948-7_21

    Google Scholar 

  8. Biryukov, A., Perrin, L.: Lightweight Cryptography Lounge (2015). http://cryptolux.org/index.php/Lightweight_Cryptography

  9. Bellare, M., Ristenpart, T., Rogaway, P., Stegers, T.: Format-preserving encryption. In: Jacobson, M.J., Rijmen, V., Safavi-Naini, R. (eds.) SAC 2009. LNCS, vol. 5867, pp. 295–312. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-05445-7_19

    Chapter  Google Scholar 

  10. Beaulieu, R., Shors, D., Smith, J., Treatman-Clark, S., Weeks, B., Wingers, L.: The Simon and Speck block ciphers on AVR 8-bit microcontrollers. In: Eisenbarth, T., Öztürk, E. (eds.) LightSec 2014. LNCS, vol. 8898, pp. 3–20. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-16363-5_1

    Google Scholar 

  11. De Cannière, C., Dunkelman, O., Knežević, M.: KATAN and KTANTAN - a family of small and efficient hardware-oriented block ciphers. In: Clavier, C., Gaj, K. (eds.) CHES 2009. LNCS, vol. 5747, pp. 272–288. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04138-9_20

    Chapter  Google Scholar 

  12. Ducas, L., Durmus, A., Lepoint, T., Lyubashevsky, V.: Lattice signatures and bimodal Gaussians. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 40–56. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_3

    Chapter  Google Scholar 

  13. Devroye, L.: Non-Uniform Random Variate Generation. Springer, Heidelberg (1986). https://doi.org/10.1007/978-1-4613-8643-8

    Book  MATH  Google Scholar 

  14. Dwarakanath, N.C., Galbraith, S.D.: Sampling from discrete Gaussians for lattice-based cryptography on a constrained device. Appl. Algebra Eng. Commun. Comput. 25(3), 159–180 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dinu, D., Perrin, L., Udovenko, A., Velichkov, V., Großschädl, J., Biryukov, A.: Design strategies for ARX with provable bounds: Sparx and LAX. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 484–513. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_18

    Chapter  Google Scholar 

  16. Fellerl, W.: An Introduction to Probability Theory and Its Applications. Wiley, London (1971)

    Google Scholar 

  17. Farach-Colton, M., Tsai, M.-T.: Exact sublinear binomial sampling. Algorithmica 73(4), 637–651 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Goldreich, O.: Modern Cryptography, Probabilistic Proofs and Pseudorandomnes. Springer, Heidelberg (1999). https://doi.org/10.1007/978-3-662-12521-2

    Book  MATH  Google Scholar 

  19. Granboulan, L., Pornin, T.: Perfect block ciphers with small blocks. In: Biryukov, A. (ed.) FSE 2007. LNCS, vol. 4593, pp. 452–465. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74619-5_28

    Chapter  Google Scholar 

  20. Guo, J., Peyrin, T., Poschmann, A., Robshaw, M.: The LED block cipher. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 326–341. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23951-9_22

    Chapter  Google Scholar 

  21. Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC, pp. 197–206. ACM (2008)

    Google Scholar 

  22. Hoang, V.T., Morris, B., Rogaway, P.: An enciphering scheme based on a card shuffle. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 1–13. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_1

    Chapter  Google Scholar 

  23. Karney, C.F.F.: Sampling exactly from the normal distribution. ACM Trans. Math. Softw. 42(1), 3:1–3:14 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lyubashevsky, V.: Lattice signatures without trapdoors. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 738–755. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_43

    Chapter  Google Scholar 

  25. Micciancio, D.: Lattice-based cryptography. In: Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.) Encyclopedia of Cryptography and Security, 2nd edn, pp. 713–715. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-540-88702-7_5

    Google Scholar 

  26. Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998)

    Article  MATH  Google Scholar 

  27. The GNU MPFR Library. http://www.mpfr.org/. Accessed 29 Sep 2017

  28. Morris, B., Rogaway, P., Stegers, T.: How to encipher messages on a small domain. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 286–302. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_17

    Chapter  Google Scholar 

  29. Peikert, C.: An efficient and parallel Gaussian sampler for lattices. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 80–97. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_5

    Chapter  Google Scholar 

  30. Rivest, R.L.: The RC5 encryption algorithm. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 86–96. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60590-8_7

    Chapter  Google Scholar 

  31. Shibutani, K., Isobe, T., Hiwatari, H., Mitsuda, A., Akishita, T., Shirai, T.: Piccolo: an ultra-lightweight blockcipher. In: Preneel, B., Takagi, T. (eds.) CHES 2011. LNCS, vol. 6917, pp. 342–357. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23951-9_23

    Chapter  Google Scholar 

  32. Suzaki, T., Minematsu, K., Morioka, S., Kobayashi, E.: \(\mathit{TWINE}\): a lightweight block cipher for multiple platforms. In: Knudsen, L.R., Wu, H. (eds.) SAC 2012. LNCS, vol. 7707, pp. 339–354. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35999-6_22

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful comments.

Author information

Authors and Affiliations

  1. NEC Corporation, Kawasaki, Japan

    Kazuhiko Minematsu, Kentarou Sasaki & Yuki Tanaka

Authors
  1. Kazuhiko Minematsu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kentarou Sasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuki Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuhiko Minematsu .

Editor information

Editors and Affiliations

  1. KU Leuven , Leuven, Belgium

    Nigel P. Smart

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Minematsu, K., Sasaki, K., Tanaka, Y. (2018). Count-then-Permute: A Precision-Free Alternative to Inversion Sampling. In: Smart, N. (eds) Topics in Cryptology – CT-RSA 2018. CT-RSA 2018. Lecture Notes in Computer Science(), vol 10808. Springer, Cham. https://doi.org/10.1007/978-3-319-76953-0_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-76953-0_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-76952-3

  • Online ISBN: 978-3-319-76953-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation