[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content

Roots of Square: Cryptanalysis of Double-Layer Square and Square+

  • Conference paper
Post-Quantum Cryptography (PQCrypto 2011)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7071))

Included in the following conference series:

Abstract

Square is a multivariate quadratic encryption scheme proposed in 2009. It is a specialization of Hidden Field Equations by using only odd characteristic fields and also X 2 as its central map. In addition, it uses embedding to reduce the number of variables in the public key. However, the system was broken at Asiacrypt 2009 using a differential attack. At PQCrypto 2010 Clough and Ding proposed two new variants named Double-Layer Square and Square+. We show how to break Double-Layer Square using a refined MinRank attack in 245 field operations. A similar fate awaits Square+ as it will be broken in 232 field operations using a mixed MinRank attack over both the extension and the ground field. Both attacks recover the private key, given access to the public key. We also outline how possible variants such as Square– or multi-Square can be attacked.

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

Access this chapter

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 35.99
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
GBP 44.99
Price includes VAT (United Kingdom)
  • 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Akkar, M.-L., Courtois, N.T., Duteuil, R., Goubin, L.: A Fast and Secure Implementation of Sflash. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 267–278. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  2. Bettale, L., Faugère, J.-C., Perret, L.: Cryptanalysis of Multivariate and Odd-Characteristic HFE Variants. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 441–458. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  3. Billet, O., Gilbert, H.: Cryptanalysis of Rainbow. In: De Prisco, R., Yung, M. (eds.) SCN 2006. LNCS, vol. 4116, pp. 336–347. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  4. Billet, O., Macario-Rat, G.: Cryptanalysis of the Square Cryptosystems. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 451–468. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  5. Buss, J.F., Frandsen, G.S., Shallit, J.O.: The computational complexity of some problems of linear algebra. Research Series RS-96-33, BRICS, Department of Computer Science, University of Aarhus, pages 39 (September 1996), http://www.brics.dk/RS/96/33/

  6. Chen, C.-H.O., Chen, M.-S., Ding, J., Werner, F., Yang, B.-Y.: Odd-char multivariate hidden field equations. Cryptology ePrint Archive, Report 2008/543 (2008), http://eprint.iacr.org/

  7. Clough, C., Baena, J., Ding, J., Yang, B.-Y., Chen, M.-S.: Square, a New Multivariate Encryption Scheme. In: Fischlin, M. (ed.) CT-RSA 2009. LNCS, vol. 5473, pp. 252–264. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  8. Clough, C.L.: Square: A New Family of Multivariate Encryption Schemes. PhD thesis, University of Cincinnati (2009)

    Google Scholar 

  9. Clough, C.L., Ding, J.: Secure Variants of the Square Encryption Scheme. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 153–164. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  10. Courtois, N., Goubin, L., Patarin, J.: Sflash: Primitive specification (second revised version) Submissions, Sflash, 11 pages (2002), https://www.cosic.esat.kuleuven.be/nessie

  11. Ding, J., Schmidt, D.: Rainbow, a New Multivariable Polynomial Signature Scheme. In: Ioannidis, J., Keromytis, A.D., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  12. Faugère, J.-C., dit Vehel, F.L., Perret, L.: Cryptanalysis of MinRank. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 280–296. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  13. Goubin, L., Courtois, N.T.: Cryptanalysis of the TTM Cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  14. Imai, H., Matsumoto, T.: Algebraic methods for constructing asymmetric cryptosystems. In: Calmet, J. (ed.) AAECC 1985. LNCS, vol. 229, pp. 108–119. Springer, Heidelberg (1986)

    Chapter  Google Scholar 

  15. Kipnis, A., Shamir, A.: Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999), http://www.minrank.org/hfesubreg.ps , http://citeseer.nj.nec.com/kipnis99cryptanalysis.html

    Chapter  Google Scholar 

  16. Matsumoto, T., Imai, H., Harashima, H., Miyakawa, H.: A cryptographically useful theorem on the connection between uni and multivariate polynomials. Transactions of the IECE of Japan 68(3), 139–146 (1985)

    Google Scholar 

  17. Patarin, J.: Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996), http://www.minrank.org/hfe.pdf

    Chapter  Google Scholar 

  18. Wolf, C., Braeken, A., Preneel, B.: Efficient Cryptanalysis of RSE(2)PKC and RSSE(2)PKC. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 294–309. Springer, Heidelberg (2005), http://eprint.iacr.org/2004/237

    Chapter  Google Scholar 

  19. Wolf, C., Braeken, A., Preneel, B.: On the security of stepwise triangular systems. Designes, Codes and Cryptography 40(3), 285–302 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wolf, C., Preneel, B.: Equivalent keys in multivariate quadratic public key systems. Journal of Mathematical Cryptology 4(4), 375–415 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang, B.-Y., Chen, J.-M.: Rank attacks and defence in Tame-like multivariate PKC’s. Cryptology ePrint Archive, Report 2004/061, September 29, pages 21 (2004), http://eprint.iacr.org/

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Thomae, E., Wolf, C. (2011). Roots of Square: Cryptanalysis of Double-Layer Square and Square+. In: Yang, BY. (eds) Post-Quantum Cryptography. PQCrypto 2011. Lecture Notes in Computer Science, vol 7071. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25405-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-25405-5_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25404-8

  • Online ISBN: 978-3-642-25405-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics