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BY-NC-ND 3.0 license Open Access Published by De Gruyter May 10, 2016

New lattice attacks on DSA schemes

  • Dimitrios Poulakis EMAIL logo

Abstract

We prove that a system of linear congruences of a particular form has at most a unique solution below a certain bound which can be computed efficiently. Using this result, we develop attacks against the DSA schemes which, under some assumptions, can provide the secret key in the case where one or several signed messages are available.

References

1 M. Bellare, S. Goldwasser and D. Micciancio, “Pseudo-random” number generation within cryptographic algorithms: The DSS case, Advances in Cryptology (CRYPTO '97), Lecture Notes in Comput. Sci. 1294, Springer, Berlin (1997), 277–291. 10.1007/BFb0052242Search in Google Scholar

2 I. F. Blake and T. Garefalakis, On the security of the digital signature algorithm, Des. Codes Cryptogr. 26 (2002), 1–3, 87–96. 10.1023/A:1016549024113Search in Google Scholar

3 I. F. Blake, G. Seroussi and N. Smart, Elliptic Curves in Cryptography, Cambridge University Press, Cambridge, 2000. 10.1017/CBO9781107360211Search in Google Scholar

4 R. Brent and P. Zimmerman, Modern Computer Arithmetic, Cambridge University Press, Cambridge, 2011. Search in Google Scholar

5 K. Draziotis and D. Poulakis, Lattice attacks on DSA schemes based on Lagrange's algorithm, Algebraic Informatics (CAI 2013), Lecture Notes in Comput. Sci. 8080, Springer, Berlin (2013), 119–131. 10.1007/978-3-642-40663-8_13Search in Google Scholar

6 T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithm, IEEE Trans. Inform. Theory 31 (1985), 469–472. 10.1007/3-540-39568-7_2Search in Google Scholar

7 J.-L. Faugère, C. Goyet and G. Renault, Attacking (EC)DSA given only an implicit hint, Selected Area of Cryptography (SAC 2012), Lecture Notes in Comput. Sci. 7707, Springer, Berlin (2013), 252–274. 10.1007/978-3-642-35999-6_17Search in Google Scholar

8 M. Girault, G. Poupard and J. Stern, Global Payment System (GPS): Un protocole de signature à la volée, Proceedings of Trusting Electronic Trade, 1999. Search in Google Scholar

9 N. A. Howgrave-Graham and N. P. Smart, Lattice Attacks on Digital Signature Schemes, Des. Codes Cryptogr. 23 (2001), 283–290. 10.1023/A:1011214926272Search in Google Scholar

10 D. Johnson, A. J. Menezes and S. A. Vastone, The elliptic curve digital signature algorithm (ECDSA), Int. J. Inf. Secur. 1 (2001), 36–63. 10.1007/s102070100002Search in Google Scholar

11 N. Koblitz and A. J. Menezes, A survey of public-key cryptosystems, SIAM Rev. 46 (2004), 4, 599–634. 10.1137/S0036144503439190Search in Google Scholar

12 N. Koblitz, A. J. Menezes and S. A. Vastone, The state of elliptic curve cryptography, Des. Codes Cryptogr. 19 (2000), 173–193. 10.1007/978-1-4757-6856-5_5Search in Google Scholar

13 A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 513–534. 10.1007/BF01457454Search in Google Scholar

14 A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, 1997. Search in Google Scholar

15 D. Micciancio and P. Voulgaris, A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations, Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC '10), Association for Computing Machinery, New York (2010), 351–358. 10.1145/1806689.1806739Search in Google Scholar

16 P. Nguyen and I. E. Shparlinski, The insecurity of the digital signature algorithm with partially known nonces, J. Cryptology 15 (2002), 151–176. 10.1007/s00145-002-0021-3Search in Google Scholar

17 P. Nguyen and I. E. Shparlinski, The insecurity of the elliptic curve digital signature algorithm with partially known nonces, Des. Codes Cryptogr. 30 (2003), 201–217. 10.1023/A:1025436905711Search in Google Scholar

18 D. Poulakis, Some Lattice Attacks on DSA and ECDSA, Appl. Algebra Engrg. Comm. Comput. 22 (2011), 347–358. 10.1007/s00200-011-0154-4Search in Google Scholar

19 D. R. Stinson, Cryptography, Theory and Practice, 2nd ed., Chapman & Hall/CRC, Boca Raton, 2001. Search in Google Scholar

20 National Institute of Standards and Technology (NIST), FIPS Publication 186: Digital Signature Standard, 1994. Search in Google Scholar

Received: 2014-7-23
Accepted: 2016-4-21
Published Online: 2016-5-10
Published in Print: 2016-6-1

© 2016 by De Gruyter

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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