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A matrix approach for FCSR automata

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Abstract

LFSRs are primitives widely used in information theory, coding theory and cryptography. However since 2002, they have faced algebraic attacks. To avoid this kind of attacks, FCSRs have been proposed as an alternative in [2–4]. In this paper, we first give a general representation of 2-adic automata using a traditional matrix representation. We then explore the special case of binary and ternary automata. We also study the complexity in terms of memory to implement such automata. Finally, we expose some proposed FCSR constructions for hardware and software oriented stream ciphers.

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References

  1. Arnault, F., Berger, T.P.: Design and properties of a new pseudorandom generator based on a filtered FCSR automaton. IEEE Trans Comput 54(11), 1374–1383 (2005)

    Article  Google Scholar 

  2. Arnault, F., Berger, T.P.: F-FCSR: design of a new class of stream ciphers. In: Gilbert, H., Handschuh, H. (eds.) FSE. Lecture Notes in Computer Science, vol. 3557, pp. 83–97. Springer, New York (2005)

    Google Scholar 

  3. Arnault, F., Berger, T.P., Lauradoux, C.: The FCSR: primitive specification and supporting documentation. ECRYPT - Network of Excellence in Cryptology. http://www.ecrypt.eu.org/stream/ (2005)

  4. Arnault, F., Berger, T.P., Lauradoux, C.: Update on F-FCSR stream cipher. ECRYPT - Network of Excellence in Cryptology. http://www.ecrypt.eu.org/stream/ (2006)

  5. Arnault, F., Berger, T.P., Lauradoux, C., Minier, M.: X-FCSR—a new software oriented stream cipher based upon FCSRs. In: Srinathan, K., Rangan, C.P., Yung, M. (eds.) INDOCRYPT. Lecture Notes in Computer Science, vol. 4859, pp. 341–350. Springer, New York (2007)

    Google Scholar 

  6. Arnault, F., Berger, T.P., Lauradoux, C., Minier, M., Pousse, B.: A new approach for FCSRs. In: M.J.J. Jr., Rijmen, V., Safavi-Naini, R., (eds.) Selected Areas in Cryptography. Lecture Notes in Computer Science, vol. 5867, pp. 433–448. Springer, New York (2009)

    Google Scholar 

  7. Arnault, F., Berger, T.P., Minier, M.: Some results on FCSR automata with applications to the security of FCSR-Based pseudorandom generators. IEEE Trans. Inf. Theory 54(2), 836–840 (2008)

    Article  MathSciNet  Google Scholar 

  8. Arnault, F., Berger, T.P., Necer, A.: Feedback with carry shift registers synthesis with the Euclidean algorithm. IEEE Trans. Inf. Theory 50(5), 910–917 (2004)

    Article  MathSciNet  Google Scholar 

  9. Berger, T.P., Minier, M., Pousse, B.: Software oriented stream ciphers based upon FCSRs in diversified mode. In: Roy, B.K., Sendrier, N. (eds.) INDOCRYPT. Lecture Notes in Computer Science, vol. 5922, pp. 119–135. Springer, New York (2009)

    Google Scholar 

  10. Ebeid, N.M., Hasan, A.: On binary signed digit representations of integers. Des. Codes Cryptography 42(1), 43–65 (2007)

    Article  MathSciNet  Google Scholar 

  11. Fischer, S., Meier, W., Stegemann, D.: Equivalent representations of the F-FCSR keystream generator. In: ECRYPT Network of Excellence—SASC Workshop, pp. 87–94. Available at http://www.ecrypt.eu.org/stvl/sasc2008/ (2008)

  12. Goresky, M., Klapper, A.: Arithmetic crosscorrelations of feedback with carry shift register sequences. IEEE Trans. Inf. Theory 43(4), 1342–1345 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Goresky, M., Klapper, A.: Fibonacci and Galois representations of feedback-with-carry shift registers. IEEE Trans. Inf. Theory 48(11), 2826–2836 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Goresky, M., Klapper, A.: Periodicity and distribution properties of combined FCSR sequences. In: Gong, G., Helleseth, T., Song, H.Y., Yang, K., (eds.) SETA. Lecture Notes in Computer Science, vol. 4086, pp. 334–341. Springer, New York (2006)

    Google Scholar 

  15. Goresky, M., Klapper, A.: Algebraic shift register sequences. Available at http://cs.engr.uky.edu/~klapper/algebraic.html (2009)

  16. Hankerson, D., Vanstone, S., Menezes, A.: Guide to Elliptic Curve Cryptography. Springer, New York (2004)

    MATH  Google Scholar 

  17. Hell, M., Johansson, T.: Breaking the F-FCSR-H Stream Cipher in Real Time. In: Pieprzyk, J. (ed.) ASIACRYPT. Lecture Notes in Computer Science, vol. 5350, pp. 557–569. Springer, New York (2008)

    Google Scholar 

  18. Joux, A., Delaunay, P.: Galois LFSR, embedded devices and side channel weaknesses. In: Progress in Cryptology—INDOCRYPT 2006. Lecture Notes in Computer Science 4329, pp. 436–451. Springer, New York (2006)

    Google Scholar 

  19. Klapper, A., Goresky, M.: 2-adic shift registers. In: Anderson, R.J. (ed.) FSE. Lecture Notes in Computer Science, vol. 809, pp. 174–178. Springer, New York (1993)

    Google Scholar 

  20. Klapper, A., Goresky, M.: Feedback shift registers, 2-adic span, and combiners with memory. J. Cryptol. 10(2), 111–147 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Klapper, A., Xu, J.: Algebraic feedback shift registers. Theor. Comput. Sci. 226(1–2), 61–92 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lauradoux, C.: Extended windmill polynomials. In: ISIT’09: Proceedings of the 2009 IEEE International Conference on Symposium on Information Theory, pp. 1120–1124. IEEE Press, Piscataway, NJ, USA (2009)

  23. Marsaglia, G.: Xorshift RNGs. J. Stat. Softw. 8(14), 1–6 (2003)

    Google Scholar 

  24. Panneton, F., L’Ecuyer, P.: On the xorshift random number generators. ACM Trans. Model. Comput. Simul. 15(4), 346–361 (2005)

    Article  Google Scholar 

  25. Smeets, B.J.M., Chambers, W.G.: Windmill generators: a generalization and an observation of how many there are. In: EUROCRYPT, pp. 325–330 (1988)

  26. Stankovski, P., Hell, M., Johansson, T.: An efficient state recovery attack on X-FCSR-256. In: Dunkelman, O. (ed.) FSE. Lecture notes in computer science, vol. 5665, pp. 23–37. Springer, New York (2009)

    Google Scholar 

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Acknowledgements

The authors would like to thank Marine Minier and the reviewers for their numerous comments that helped to improve this paper.

This work was partially supported by the French National Agency of Research: ANR-06-SETI-013.

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Authors and Affiliations

  1. XLIM (UMR CNRS 6172), Université de Limoges, 123 avenue Albert Thomas, 87060, Limoges Cedex, France

    François Arnault, Thierry P. Berger & Benjamin Pousse

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  1. François Arnault
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  2. Thierry P. Berger
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  3. Benjamin Pousse
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Correspondence to Benjamin Pousse.

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Arnault, F., Berger, T.P. & Pousse, B. A matrix approach for FCSR automata. Cryptogr. Commun. 3, 109–139 (2011). https://doi.org/10.1007/s12095-010-0041-z

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  • DOI: https://doi.org/10.1007/s12095-010-0041-z

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