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.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
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)
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)
Babai, L.: On Lovász lattice reduction and the nearest lattice point problem. Combinatorica 6, 1–13 (1986)
Bard, G.V.: Algebraic Cryptanalysis. Springer, Heidelberg (2001) ISBN 978-0-387-88756-2
Bosma, W., Cannon, J., Playoust, C.: The Magma Algebra System I: The user language. Journal of Symbolic Computation 24, 235–265 (1997)
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)
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progression. Journal of Symbolic Computation 9, 251–280 (1990)
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)
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)
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)
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
Goldwasser, S., Micali, S.: Probabilistic Encryption. J. Computer and System Sciences 28, 270–299 (1983)
Håstad, J.: Solving simultaneous modular equations of low degree. SIAM J. Comput. 17, 336–341 (1988)
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)
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)
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
Howgrave-Graham, N., Silverman, J.H.: Implementation Notes for NTRU PKCS Multiple Transmissions. Technical report, http://www.ntru.com/cryptolab/technotes.htm#006
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)
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)
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)
Odlyzko, A.M.: The rise and fall of knapsack cryptosystems. Cryptology and Computational Number Theory 42, 75–88 (1990)
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)
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)
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)
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
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)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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)