Abstract
A public key cryptosystem based on factoring and a combinatorial problem of matrices over ℤ N proposed in 2010 is analyzed in this paper. We propose an efficient partial private key recovery attack on it by solving a problem of recovering implicit polynomials with small coefficients given their large roots and deriving the large roots from the public key. From the partial information of private key, we can decrypt any ciphertext of the cryptosystem by a simple computation. Our implicit polynomial recovery is an application of lattice basis reduction.
Chapter PDF
Similar content being viewed by others
Keywords
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)
Bosma, W., Cannon, J., Playoust, C.: The Magma Algebra System I: The user language. Journal of Symbolic Computation 24, 235–265 (1997)
Coppersmith, D.: Finding a Small Root of a Univariate Modular Equation. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 155–165. Springer, Heidelberg (1996)
Coppersmith, D., Franklin, M., Patarin, J., Reiter, M.: Low-Exponent RSA with Related Messages. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 1–9. Springer, Heidelberg (1996)
Goldreich, O., Goldwasser, S., Halvei, S.: Public-Key Cryptosystems from Lattice Reduction Problems. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 112–131. Springer, Heidelberg (1997)
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)
Lenstra, A.K., Lenstra Jr., H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)
Merkle, R.C., Hellman, M.E.: Hiding Information and Signatures in Trapdoor Knapsack. IEEE Transaction on Information Theory 24, 525–530 (1978)
Nguyen, 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)
Shoup, V.: A library for doing number theory, http://www.shoup.net/ntl
Wang, B., Hu, Y.: Diophantine Approximation Attack on a Fast Public Key Cryptosystem. In: Chen, K., Deng, R., Lai, X., Zhou, J. (eds.) ISPEC 2006. LNCS, vol. 3903, pp. 25–32. Springer, Heidelberg (2006)
Wang, B., Hu, Y.: A Novel Combinatorial Public Key Cryptosystem. Informatica 21(4), 611–626 (2010)
Zwillinger, D.(editor in chief): CRC Standard Mathematical Tables and Formulae, 30th edn. CRC Press, Boca Raton (1996)
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. (2012). Implicit Polynomial Recovery and Cryptanalysis of a Combinatorial Key Cryptosystem. In: Chim, T.W., Yuen, T.H. (eds) Information and Communications Security. ICICS 2012. Lecture Notes in Computer Science, vol 7618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34129-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-34129-8_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34128-1
Online ISBN: 978-3-642-34129-8
eBook Packages: Computer ScienceComputer Science (R0)