Abstract
Stream ciphers efficiently encrypt data streams of arbitrary length and are widely deployed in practice, e.g., in mobile phones. Consequently, the development of new mechanisms to design and analyze stream ciphers is one of the major topics in modern cryptography. Algebraic attacks evaluate the security of certain stream ciphers by exploring the question how an attack could be performed by generating and solving appropriate systems of equations. In this text, we give an introduction to algebraic attacks and provide an overview on how and to what extent Gröbner bases are useful in this context.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Armknecht, Improving fast algebraic attacks, Proc. of FSE 2004, LNCS, vol. 3017, Springer, Berlin, 2004a, pp. 65–82.
F. Armknecht, On the existence of low-degree equations for algebraic attacks, Cryptology ePrint Archive, Report 2004/185, 2004b, http://eprint.iacr.org/.
F. Armknecht, Algebraic attacks and annihilators, Proc. of WEWORC 2005, LNI, vol. 74, 2005, pp. 13–21.
F. Armknecht, Algebraic attacks on certain stream ciphers, Ph.D. thesis, University Mannheim, Germany, 2006.
F. Armknecht and G. Ars, Introducing a new variant of fast algebraic attacks and minimizing their successive data complexity, Proc. of Mycrypt, LNCS, vol. 3715, Springer, Berlin, 2005, pp. 16–32.
F. Armknecht and M. Krause, Algebraic attacks on combiners with memory, Proc. of CRYPTO 2003, LNCS, vol. 2729, 2003, pp. 162–175.
F. Armknecht, C. Carlet, P. Gaborit, S. Künzli, W. Meier, and O. Ruatta, Efficient computation of algebraic immunity for algebraic and fast algebraic attacks, Proc. of Eurocrypt 2006, LNCS, vol. 4004, 2006, pp. 147–164.
G. Ars, Applications of Gröbner Bases to Cryptography, Ph.D. thesis, University of Rennes I, 2005.
G. Ars and J. C. Faugère, An algebraic cryptanalysis of nonlinear filter generators using Gröbner bases, INRIA Report 4739, 2003, http://www.inria.fr/rrrt/rr-4739.html.
G. Ars and J. C. Faugère, Algebraic immunities of functions over finite fields, Tech. report, INRIA, 2005, ftp://ftp.inria.fr/INRIA/publication.
M. Bardet, J. C. Faugere, B. Salvy, and B. Y. Yang, Asymptotic behaviour of the degree of regularity of semi-regular polynomial systems, Tech. report, Talk at MEGA 2005, 2005.
Bluetooth specification v1.1, 1999, http://www.bluetooth.com/.
M. Briceno, I. Goldberg, and D. Wagner, A pedagogical implementation of A5/1, 1998, http://jya.com/a51-pi.htm.
B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis, Innsbruck, 1965.
B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math. 4 (1970), 374–383.
B. Buchberger, Gröbner-bases: An algorithmic method in polynomial ideal theory, Multidimensional systems theory, Reidel, Dordrecht, 1985, pp. 184–232.
B. Buchberger, An algorithmical criterion for the solvability of algebraic systems of equations, London Math. Soc. LNS 251 (1998), 535–545.
B. Buchberger, Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symb. Comput. 41 (2006), nos. 3–4, 475–511.
A. Canteaut and E. Filiol, Ciphertext only reconstruction of stream ciphers based on combination generators, Proc. of FSE 2000, LNCS, vol. 1978, Springer, Berlin, 2000, pp. 165–180.
A. Canteaut and E. Filiol, On the influence of the filtering function on the performance of fast correlation attacks on filter generators, Proc. of Symposium on Information Theory 2002, 2002.
J. Cho and J. Pieprzyk, Algebraic attacks on SOBER-t32 and SOBER-t16 without stuttering, Proc. of FSE 2004, LNCS, vol. 3017, Springer, Berlin, 2004, pp. 49–64.
N. Courtois, Fast algebraic attacks on stream ciphers with linear feedback, Proc. of CRYPTO 2003, LNCS, vol. 2656, Springer, Berlin, 2003, pp. 176–194.
N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, Proc. of EUROCRYPT 2003, LNCS, vol. 2656, Springer, Berlin, 2003, pp. 345–359.
F. Didier and J. Tillich, Computing the algebraic immunity efficiently, Proc. of FSE 2006, LNCS, vol. 4047, Springer, Berlin, 2006, pp. 359–374.
J. C. Faugère, A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5), Proc. of ISSAC 2002, ACM, New York, 2002, pp. 75–83.
S. Fluhrer, I. Mantin, and A. Shamir, Weaknesses in the key scheduling algorithm of RC4, Proc. of SAC 2001, Springer, Berlin, 2001, pp. 1–24.
P. Geffe, How to protect data with ciphers that are really hard to break, Electronics 46 (1973), no. 1, 99–101.
J. Håstad, S. Phillips, and S. Safra, A well-characterized approximation problem, Inf. Process. Lett. 47 (1993), no. 6, 301–305.
P. Hawkes and G. Rose, Rewriting variables: The complexity of fast algebraic attacks on stream ciphers, Proc. of CRYPTO 2004, LNCS, vol. 3152, Springer, Berlin, 2004, pp. 390–406.
A. Kerckhoffs, La cryptographie militaire, Journal des Sciences Militaires (1883), 161–191.
D. Lee, J. Kim, J. Hong, J. Han, and D. Moon, Algebraic attacks on summation generators, Proc. of FSE2004, LNCS, vol. 3017, Springer, Berlin, 2004, pp. 34–48.
R. Lidl and H. Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, Cambridge, 1986.
W. Meier, E. Pasalic, and C. Carlet, Algebraic attacks and decomposition of Boolean functions, Proc. of EUROCRYPT 2004, LNCS, vol. 3027, Springer, Berlin, 2004, pp. 474–491.
T. Mora, Gröbner technology, this volume, 2009, pp. 11–25.
R. Rueppel, Correlation immunity and the summation generator, Proc. of CRYPTO 1985, LNCS, vol. 218, Springer, Berlin, 1985, pp. 260–272.
R. Rueppel, Security models and notions for stream ciphers, Proc. of 2nd IMA Conference on Cryptography and Coding, Oxford University Press, London, 1989, pp. 213–230.
R. Rueppel, Stream ciphers, Contemporary cryptology—The science of information integrity, IEEE Press, 1992, pp. 65–134.
A. Seidenberg, Constructions in algebra, Trans. Amer. Math. Soc. 197 (1974), 273–313.
C. E. Shannon, Communication theory of secrecy systems, Bell System Tech. J. 28 (1949), 656–715.
V. Strassen, Gaussian elimination is not optimal, Numerische Mathematik 13 (1969), 354–356.
S. Wolfram, Random sequence generation by cellular automata, Advances in Applied Mathematics 7 (1986), 123–169.
E. Zenner, On cryptographic properties of LFSR-based pseudorandom generators, Ph.D. thesis, Universität Mannheim, 2004.
E. Zenner, R. Weis, and S. Lucks, Sicherheit des GSM-Verschlüsselungsstandards A5, Datenschutz und Datensicherheit 24 (2000), no. 7, 405–407.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Armknecht, F., Ars, G. (2009). Algebraic Attacks on Stream Ciphers with Gröbner Bases. In: Sala, M., Sakata, S., Mora, T., Traverso, C., Perret, L. (eds) Gröbner Bases, Coding, and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93806-4_18
Download citation
DOI: https://doi.org/10.1007/978-3-540-93806-4_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-93805-7
Online ISBN: 978-3-540-93806-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)