Skip to main content
SpringerLink
Log in
Book cover

International Conference on Cryptology and Network Security

CANS 2012: Cryptology and Network Security pp 32–42Cite as

Cryptanalysis of a Lattice-Knapsack Mixed Public Key Cryptosystem

  • Conference paper

  • 1244 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7712))

Abstract

Recently, a lattice based public key cryptosystem mixed with a knapsack was presented in the CANS 2011 conference. In this paper, we propose two message recovery attacks on this cryptosystem. The first one is a broadcast attack: a single message of m bits can be recovered if it is encrypted for \(\lceil\frac{m+1}{2}\rceil\) recipients. The second attack is a multiple transmission attack in which a message can be recovered with a probability of (1 − 2− l)m if it is encrypted under a same public key for l = ⌈log2 m + 2⌉ times using different random numbers. The multiple transmission attack can be further improved with a linearization technique to that only \(\lceil\frac{\log_2m+1}{2}\rceil\) times of encryptions are required to recover the message. An open problem related to the message recovery attack using only one cipehertext is discussed.

This is a preview of subscription content, 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 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pp. 284–293 (1997)

    Google Scholar 

  2. Arora, S., Ge, R.: New Algorithms for Learning in Presence of Errors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 403–415. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  3. Babai, L.: On Lovász lattice reduction and the nearest lattice point problem. Combinatorica 6, 1–13 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bard, G.V.: Algebraic Cryptanalysis. Springer, Heidelberg (2001) ISBN 978-0-387-88756-2

    Google Scholar 

  5. Bosma, W., Cannon, J., Playoust, C.: The Magma Algebra System I: The user language. Journal of Symbolic Computation 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cai, J.-Y., Cusick, T.W.: A Lattice-Based Public-Key Cryptosystem. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 219–233. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  7. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progression. Journal of Symbolic Computation 9, 251–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Courtois, N.T., Bard, G.V.: Algebraic Cryptanalysis of the Data Encryption Standard. In: Galbraith, S.D. (ed.) Cryptography and Coding 2007. LNCS, vol. 4887, pp. 152–169. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  9. Courtois, N.T., Klimov, A., Patarin, J., Shamir, A.: Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  10. Ding, J., Hu, L., Nie, X., Li, J., Wagner, J.: High Order Linearization Equation (HOLE) Attack on Multivariate Public Key Cryptosystems. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 233–248. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  11. Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, pp. 197–206. ACM Press (2008) ISBN 978-1-60558-047-0

    Google Scholar 

  12. Goldwasser, S., Micali, S.: Probabilistic Encryption. J. Computer and System Sciences 28, 270–299 (1983)

    Article  MathSciNet  Google Scholar 

  13. Håstad, J.: Solving simultaneous modular equations of low degree. SIAM J. Comput. 17, 336–341 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Chapter  Google Scholar 

  15. Howgrave-Graham, N.: A Hybrid Lattice-Reduction and Meet-in-the-Middle Attack Against NTRU. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 150–169. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  16. Howgrave-Graham, N., Silverman, J.H.: A Meet-In-The-Meddle Attack on an NTRU Private Key. Technical report, http://www.ntru.com/cryptolab/technotes.htm#004

  17. Howgrave-Graham, N., Silverman, J.H.: Implementation Notes for NTRU PKCS Multiple Transmissions. Technical report, http://www.ntru.com/cryptolab/technotes.htm#006

  18. Lyubashevsky, V., Peikert, C., Regev, O.: On Ideal Lattices and Learning with Errors over Rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  19. May, A., Silverman, J.H.: Dimension Reduction Methods for Convolution Modular Lattices. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 110–125. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  20. Nguyên, P.Q., Stern, J.: Cryptanalysis of the Ajtai-Dwork Cryptosystem. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 223–242. Springer, Heidelberg (1998)

    Google Scholar 

  21. Odlyzko, A.M.: The rise and fall of knapsack cryptosystems. Cryptology and Computational Number Theory 42, 75–88 (1990)

    MathSciNet  Google Scholar 

  22. Plantard, T., Susilo, W.: Broadcast Attacks against Lattice-Based Cryptosystems. In: Abdalla, M., Pointcheval, D., Fouque, P.-A., Vergnaud, D. (eds.) ACNS 2009. LNCS, vol. 5536, pp. 456–472. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  23. Pan, Y., Deng, Y.: A Ciphertext-Only Attack Against the Cai-Cusick Lattice-Based Public-Key Cryptosystem. IEEE Transactions on Information Theory 57, 1780–1785 (2011)

    Article  MathSciNet  Google Scholar 

  24. Pan, Y., Deng, Y., Jiang, Y., Tu, Z.: A New Lattice-Based Public-Key Cryptosystem Mixed with a Knapsack. In: Lin, D., Tsudik, G., Wang, X. (eds.) CANS 2011. LNCS, vol. 7092, pp. 126–137. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  25. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: The 37th Annual ACM Symposium on Theory of Computing, pp. 84–93. ACM Press (2004) ISBN 1-58113-960-8

    Google Scholar 

  26. Shor, P.: Algorithms for Quantum Computation: Discrete Logrithms and Factoring. In: The 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. IEEE Computer Science Press, Santa Fe (1994)

    Chapter  Google Scholar 

  27. Stehlé, D., Steinfeld, R.: Making NTRU as Secure as Worst-Case Problems over Ideal Lattices. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 27–47. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Jun Xu, Lei Hu & Siwei Sun

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    Jun Xu

  3. School of Mathematical Science, Anhui University, Hefei, 230601, Anhui, China

    Jun Xu

  4. Tian Jin Zhong Wei Aerospace Data System Technology Co., Ltd, China

    Ping Wang

  5. Space Star Technology Co., Ltd, China

    Ping Wang

  6. Institute No.503 of the fifth Research Academy, China Aerospace Science and Technology Corporation, Beijing, 100086, China

    Ping Wang

Authors
  1. Jun Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lei Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Siwei Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computing, Macquarie University, 2109, Sydney, NSW, Australia

    Josef Pieprzyk

  2. System Security Lab., Technische Universität Darmstadt, Darmstadt, Germany

    Ahmad-Reza Sadeghi

  3. Department of Computing, University of Surrey, GU2 7XH, Guildford, UK

    Mark Manulis

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Xu, J., Hu, L., Sun, S., Wang, P. (2012). Cryptanalysis of a Lattice-Knapsack Mixed Public Key Cryptosystem. In: Pieprzyk, J., Sadeghi, AR., Manulis, M. (eds) Cryptology and Network Security. CANS 2012. Lecture Notes in Computer Science, vol 7712. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35404-5_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-35404-5_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35403-8

  • Online ISBN: 978-3-642-35404-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation