Linear Complexity of Binary Sequences Derived from Polynomial Quotients

  • Zhixiong Chen
  • Domingo Gómez-Pérez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7280)

Abstract

We determine the linear complexity of p2-periodic binary threshold sequences derived from polynomial quotient, which is defined by the function \((u^w-u^{wp})/p \pmod p\). When w = (p − 1)/2 and \(2^{p-1}\not\equiv 1 \pmod{p^2}\), we show that the linear complexity is equal to one of the following values \(\left\{p^2-1,\ p^2-p,\ (p^2+p)/2+1,\ (p^2-p)/2\right \}\), depending whether \(p\equiv 1,\ -1,\ 3,\ -3\pmod 8\). But it seems that the method can’t be applied to the case of general w.

Keywords

Fermat quotients polynomial quotients finite fields pseudorandom binary sequences linear complexity cryptography 

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References

  1. 1.
    Ernvall, R., Metsänkylä, T.: On the p-divisibility of Fermat quotients. Math. Comp. 66(219), 1353–1365 (1997)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Granville, A.: Some conjectures related to Fermat’s last theorem. In: Number Theory (Banff, AB, 1988), pp. 177–192. de Gruyter, Berlin (1990)Google Scholar
  3. 3.
    Chang, M.C.: Short character sums with Fermat quotients. Acta Arith. 152(1), 23–38 (2012)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Chen, Z., Winterhof, A.: Additive character sums of polynomial quotients (preprint)Google Scholar
  5. 5.
    Shparlinski, I.: Character sums with Fermat quotients. Quart. J. Math. Oxford 62(4), 1031–1043 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Shparlinski, I.E.: Bounds of multiplicative character sums with Fermat quotients of primes. Bull. Aust. Math. Soc. 83(3), 456–462 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Shparlinski, I.E.: On the value set of Fermat quotients. Proc. Amer. Math. Soc. 140(140), 1199–1206 (2011)MathSciNetGoogle Scholar
  8. 8.
    Shparlinski, I.E.: Fermat quotients: exponential sums, value set and primitive roots. Bull. Lond. Math. Soc. 43(6), 1228–1238 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chen, Z., Du, X.: On the linear complexity of binary threshold sequences derived from Fermat quotients. Des. Codes Cryptogr. (in press)Google Scholar
  10. 10.
    Chen, Z., Hu, L., Du, X.: Linear complexity of some binary sequences derived from Fermat quotients. China Communications 9(2), 105–108 (2012)Google Scholar
  11. 11.
    Chen, Z., Ostafe, A., Winterhof, A.: Structure of Pseudorandom Numbers Derived from Fermat Quotients. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. LNCS, vol. 6087, pp. 73–85. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Du, X., Klapper, A., Chen, Z.: Linear complexity of pseudorandom sequences generated by Fermat quotients and their generalizations. Inf. Proc. Letters 112(6), 233–237 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gomez, D., Winterhof, A.: Multiplicative character sums of fermat quotients and pseudorandom sequences. Period. Math. Hungar. (in press)Google Scholar
  14. 14.
    Ostafe, A., Shparlinski, I.E.: Pseudorandomness and dynamics of Fermat quotients. SIAM J. Discrete Math. 25(1), 50–71 (2011)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Lidl, R., Niederreiter, H.: Finite fields, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1997)Google Scholar
  16. 16.
    Meidl, W., Niederreiter, H.: Linear complexity, k-error linear complexity, and the discrete Fourier transform. J. Complexity 18(1), 87–103 (2002)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Winterhof, A.: A note on the linear complexity profile of the discrete logarithm in finite fields. In: Coding, Cryptography and Combinatorics. Progr. Comput. Sci. Appl. Logic, vol. 23, pp. 359–367. Birkhäuser, Basel (2004)CrossRefGoogle Scholar
  18. 18.
    Crandall, R., Dilcher, K., Pomerance, C.: A search for Wieferich and Wilson primes. Math. Comp. 66(217), 433–449 (1997)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Zhixiong Chen
    • 1
    • 2
  • Domingo Gómez-Pérez
    • 3
  1. 1.Department of MathematicsPutian UniversityPutianP.R. China
  2. 2.State Key Laboratory of Information Security, Institute of SoftwareChinese Academy of SciencesBeijingP.R. China
  3. 3.University of CantabriaSantanderSpain

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