Skip to main content
SpringerLink
Log in

A Secure Three-Input AND Protocol with a Standard Deck of Minimal Cards

  • Conference paper
  • First Online:
Computer Science – Theory and Applications (CSR 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12730))

Included in the following conference series:

Abstract

Card-based protocols are used to perform cryptographic tasks such as secure multiparty computation using a deck of physical cards. While most of the existing protocols use a two-colored deck consisting of red cards and black cards, Niemi and Renvall in 1999 constructed protocols for securely computing two-input Boolean functions (such as secure logical AND and XOR computations) using a commonly available standard deck of playing cards. Since this initial investigation, two-input protocols with fewer cards and/or shuffles have been designed, and by combining them, one can perform a secure computation of any Boolean circuit. In this paper, we directly construct a simple card-based protocol for the three-input AND computation. Our three-input AND protocol requires fewer cards and shuffles compared to that required when applying any existing two-input AND protocol twice to perform the three-input AND computation. Our protocol is unique in the sense that it is card minimal if we use two cards to encode a single bit.

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

Similar content being viewed by others

Notes

  1. 1.

    There are other types of cards used for secure computations, such as polarizing cards [29], polygon cards [30], triangle cards [28], and dihedral cards [27].

  2. 2.

    Niemi and Renvall [18] also provided Las Vegas protocols for the secure computations of XOR and copy.

  3. 3.

    Mizuki [10] also presented finite XOR and copy protocols.

References

  1. Abe, Y., Hayashi, Y., Mizuki, T., Sone, H.: Five-card AND computations in committed format using only uniform cyclic shuffles. New Gener. Comput. 39(1), 97–114 (2021). https://doi.org/10.1007/s00354-020-00110-2

    Article  Google Scholar 

  2. Boer, B.: More efficient match-making and satisfiability the five card trick. In: Quisquater, J.-J., Vandewalle, J. (eds.) EUROCRYPT 1989. LNCS, vol. 434, pp. 208–217. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-46885-4_23

    Chapter  Google Scholar 

  3. Bultel, X., Dreier, J., Dumas, J.G., Lafourcade, P.: Physical zero-knowledge proofs for Akari, Takuzu, Kakuro and KenKen. In: Demaine, E.D., Grandoni, F. (eds.) 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), vol. 49, pp. 8:1–8:20. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2016). https://doi.org/10.4230/LIPIcs.FUN.2016.8

  4. Crépeau, C., Kilian, J.: Discreet solitary games. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 319–330. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-48329-2_27

    Chapter  Google Scholar 

  5. Kastner, J., et al.: The minimum number of cards in practical card-based protocols. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10626, pp. 126–155. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70700-6_5

    Chapter  Google Scholar 

  6. Koch, A., Schrempp, M., Kirsten, M.: Card-based cryptography meets formal verification. New Gener. Comput. 39(1), 115–158 (2021). https://doi.org/10.1007/s00354-020-00120-0

    Article  MATH  Google Scholar 

  7. Koch, A., Walzer, S., Härtel, K.: Card-based cryptographic protocols using a minimal number of cards. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 783–807. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48797-6_32

    Chapter  Google Scholar 

  8. Manabe, Y., Ono, H.: Secure card-based cryptographic protocols using private operations against malicious players. In: Maimut, D., Oprina, A.-G., Sauveron, D. (eds.) SecITC 2020. LNCS, vol. 12596, pp. 55–70. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-69255-1_5

    Chapter  Google Scholar 

  9. Miyahara, D., Hayashi, Y., Mizuki, T., Sone, H.: Practical card-based implementations of Yao’s millionaire protocol. Theor. Comput. Sci. 803, 207–221 (2020). https://doi.org/10.1016/j.tcs.2019.11.005

    Article  MathSciNet  MATH  Google Scholar 

  10. Mizuki, T.: Efficient and secure multiparty computations using a standard deck of playing cards. In: Foresti, S., Persiano, G. (eds.) CANS 2016. LNCS, vol. 10052, pp. 484–499. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-48965-0_29

    Chapter  MATH  Google Scholar 

  11. Mizuki, T., Asiedu, I.K., Sone, H.: Voting with a logarithmic number of cards. In: Mauri, G., Dennunzio, A., Manzoni, L., Porreca, A.E. (eds.) UCNC 2013. LNCS, vol. 7956, pp. 162–173. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39074-6_16

    Chapter  Google Scholar 

  12. Mizuki, T., Komano, Y.: Analysis of information leakage due to operative errors in card-based protocols. In: Iliopoulos, C., Leong, H.W., Sung, W.-K. (eds.) IWOCA 2018. LNCS, vol. 10979, pp. 250–262. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94667-2_21

    Chapter  Google Scholar 

  13. Mizuki, T., Shizuya, H.: A formalization of card-based cryptographic protocols via abstract machine. Int. J. Inf. Secur. 13(1), 15–23 (2014). https://doi.org/10.1007/s10207-013-0219-4

    Article  Google Scholar 

  14. Mizuki, T., Shizuya, H.: Practical card-based cryptography. In: Ferro, A., Luccio, F., Widmayer, P. (eds.) Fun with Algorithms. LNCS, vol. 8496, pp. 313–324. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-07890-8_27

    Chapter  Google Scholar 

  15. Mizuki, T., Shizuya, H.: Computational model of card-based cryptographic protocols and its applications. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 100(1), 3–11 (2017). https://doi.org/10.1587/transfun.E100.A.3

    Article  Google Scholar 

  16. Mizuki, T., Sone, H.: Six-card secure AND and four-card secure XOR. In: Deng, X., Hopcroft, J.E., Xue, J. (eds.) Frontiers in Algorithmics. LNCS, vol. 5598, pp. 358–369. Springer, Berlin, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02270-8_36

    Chapter  Google Scholar 

  17. Nakai, T., Misawa, Y., Tokushige, Y., Iwamoto, M., Ohta, K.: How to solve millionaires’ problem with two kinds of cards. New Gener. Comput. 39(1), 73–96 (2021). https://doi.org/10.1007/s00354-020-00118-8

    Article  Google Scholar 

  18. Niemi, V., Renvall, A.: Solitaire zero-knowledge. Fundam. Inf. 38(1,2), 181–188 (1999). https://doi.org/10.3233/FI-1999-381214

    Article  MathSciNet  MATH  Google Scholar 

  19. Nishida, T., Hayashi, Y., Mizuki, T., Sone, H.: Card-based protocols for any Boolean function. In: Jain, R., Jain, S., Stephan, F. (eds.) TAMC 2015. LNCS, vol. 9076, pp. 110–121. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-17142-5_11

    Chapter  Google Scholar 

  20. Nishida, T., Hayashi, Y., Mizuki, T., Sone, H.: Securely computing three-input functions with eight cards. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 98(6), 1145–1152 (2015). https://doi.org/10.1587/transfun.E98.A.1145

    Article  Google Scholar 

  21. Nishida, T., Mizuki, T., Sone, H.: Securely computing the three-input majority function with eight cards. In: Dediu, A.-H., Martín-Vide, C., Truthe, B., Vega-Rodríguez, M.A. (eds.) TPNC 2013. LNCS, vol. 8273, pp. 193–204. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-45008-2_16

    Chapter  Google Scholar 

  22. Ono, H., Manabe, Y.: Card-based cryptographic logical computations using private operations. New Gener. Comput. 39(1), 19–40 (2020). https://doi.org/10.1007/s00354-020-00113-z

    Article  Google Scholar 

  23. Robert, L., Miyahara, D., Lafourcade, P., Mizuki, T.: Physical zero-knowledge proof for Suguru puzzle. In: Devismes, S., Mittal, N. (eds.) SSS 2020. LNCS, vol. 12514, pp. 235–247. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64348-5_19

    Chapter  Google Scholar 

  24. Ruangwises, S., Itoh, T.: AND protocols using only uniform shuffles. In: van Bevern, R., Kucherov, G. (eds.) CSR 2019. LNCS, vol. 11532, pp. 349–358. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-19955-5_30

    Chapter  MATH  Google Scholar 

  25. Ruangwises, S., Itoh, T.: Physical zero-knowledge proof for numberlink puzzle and k vertex-disjoint paths problem. New Gener. Comput. 39(1), 3–17 (2020). https://doi.org/10.1007/s00354-020-00114-y

    Article  Google Scholar 

  26. Ruangwises, S., Itoh, T.: Physical zero-knowledge proof for ripple effect. In: Uehara, R., Hong, S.-H., Nandy, S.C. (eds.) WALCOM 2021. LNCS, vol. 12635, pp. 296–307. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-68211-8_24

    Chapter  Google Scholar 

  27. Shinagawa, K.: Card-based cryptography with dihedral symmetry. New Gener. Comput. 39(1), 41–71 (2021). https://doi.org/10.1007/s00354-020-00117-9

    Article  Google Scholar 

  28. Shinagawa, K., Mizuki, T.: Card-based protocols using triangle cards. In: Ito, H., Leonardi, S., Pagli, L., Prencipe, G. (eds.) 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 100, pp. 31:1–31:13. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2018). https://doi.org/10.4230/LIPIcs.FUN.2018.31

  29. Shinagawa, K., et al.: Secure computation protocols using polarizing cards. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 99(6), 1122–1131 (2016)

    Article  Google Scholar 

  30. Shinagawa, K., et al.: Card-based protocols using regular polygon cards. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 100(9), 1900–1909 (2017)

    Article  Google Scholar 

  31. Ueda, I., Miyahara, D., Nishimura, A., Hayashi, Y., Mizuki, T., Sone, H.: Secure implementations of a random bisection cut. Int. J. Inf. Secur. 19, 445–452 (2020). https://doi.org/10.1007/s10207-019-00463-w

    Article  Google Scholar 

  32. Ueda, I., Nishimura, A., Hayashi, Y., Mizuki, T., Sone, H.: How to implement a random bisection cut. In: Martín-Vide, C., Mizuki, T., Vega-Rodríguez, M.A. (eds.) TPNC 2016. LNCS, vol. 10071, pp. 58–69. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-49001-4_5

    Chapter  Google Scholar 

  33. Yasunaga, K.: Practical card-based protocol for three-input majority. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 103(11), 1296–1298 (2020). https://doi.org/10.1587/transfun.2020EAL2025

    Article  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referees, whose comments have helped us to improve the presentation of the paper. This work was supported in part by JSPS KAKENHI Grant Number JP19J21153.

Author information

Authors and Affiliations

  1. School of Engineering, Tohoku University, 6-6 Aramaki-Aza-Aoba, Aoba-ku, Sendai, 980-8579, Japan

    Hiroto Koyama

  2. Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai, 980–8579, Japan

    Daiki Miyahara

  3. Cyberscience Center, Tohoku University, 6-3 Aramaki-Aza-Aoba, Aoba-ku, Sendai, 980-8578, Japan

    Takaaki Mizuki & Hideaki Sone

  4. National Institute of Advanced Industrial Science and Technology (AIST), 2-4-7 Aomi, Koto-ku, Tokyo, 135-0064, Japan

    Daiki Miyahara & Takaaki Mizuki

Authors
  1. Hiroto Koyama
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daiki Miyahara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Takaaki Mizuki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hideaki Sone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daiki Miyahara .

Editor information

Editors and Affiliations

  1. University of Oxford, Oxford, UK

    Rahul Santhanam

  2. Innovations & HighTech, Moscow Institute of Physics and Technology, Moscow, Russia

    Daniil Musatov

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Koyama, H., Miyahara, D., Mizuki, T., Sone, H. (2021). A Secure Three-Input AND Protocol with a Standard Deck of Minimal Cards. In: Santhanam, R., Musatov, D. (eds) Computer Science – Theory and Applications. CSR 2021. Lecture Notes in Computer Science(), vol 12730. Springer, Cham. https://doi.org/10.1007/978-3-030-79416-3_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-79416-3_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-79415-6

  • Online ISBN: 978-3-030-79416-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation