Skip to main content
SpringerLink
Log in

Structure of Pseudorandom Numbers Derived from Fermat Quotients

  • Conference paper
Arithmetic of Finite Fields (WAIFI 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6087))

Included in the following conference series:

Abstract

We study the distribution of s-dimensional points of Fermat quotients modulo p with arbitrary lags. If no lags coincide modulo p the same technique as in [21] works. However, there are some interesting twists in the other case. We prove a discrepancy bound which is unconditional for s = 2 and needs restrictions on the lags for s > 2. We apply this bound to derive results on the pseudorandomness of the binary threshold sequence derived from Fermat quotients in terms of bounds on the well-distribution measure and the correlation measure of order 2, both introduced by Mauduit and Sárközy. We also prove a lower bound on its linear complexity profile. The proofs are based on bounds on exponential sums and earlier relations between discrepancy and both measures above shown by Mauduit, Niederreiter and Sárközy. Moreover, we analyze the lattice structure of Fermat quotients modulo p with arbitrary lags.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brandstätter, N., Winterhof, A.: Linear complexity profile of binary sequences with small correlation measure. Periodica Mathematica Hungarica 52(2), 1–8 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cassaigne, J., Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences, VII: the measures of pseudorandomness. Acta Arithmetica 103, 97–118 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dorfer, G.: Lattice profile and linear complexity profile of pseudorandom number sequences. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds.) Fq7 2003. LNCS, vol. 2948, pp. 69–78. Springer, Heidelberg (2004)

    Google Scholar 

  4. Dorfer, G., Meidl, W., Winterhof, A.: Counting functions and expected values for the lattice profile at n. Finite Fields Appl. 10, 636–652 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dorfer, G., Winterhof, A.: Lattice structure and linear complexity profile of nonlinear pseudorandom number generators. Appl. Algebra Engrg. Comm. Comput. 13, 499–508 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dorfer, G., Winterhof, A.: Lattice structure of nonlinear pseudorandom number generators in parts of the period. In: Niederreiter, H. (ed.) Monte Carlo and Quasi-Monte Carlo Methods 2002, pp. 199–211. Springer, Berlin (2004)

    Google Scholar 

  7. Drmota, M., Tichy, R.: Sequences, Discrepancies and Applications. Springer, Berlin (1997)

    MATH  Google Scholar 

  8. Ernvall, R., Metsänkylä, T.: On the p-divisibility of Fermat quotients. Math. Comp. 66, 1353–1365 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fu, F.-W., Niederreiter, H.: On the counting function of the lattice profile of periodic sequences. J. Complexity 23, 423–435 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Granville, A.: Some conjectures related to Fermat’s Last Theorem. In: Number Theory W. de Gruyter, NY, pp. 177–192 (1990)

    Google Scholar 

  11. Heath-Brown, R.: An estimate for Heilbronn’s exponential sum. In: Analytic Number Theory: Proc. Conf. in honor of Heini Halberstam, Birkhäuser, Boston, pp. 451–463 (1996)

    Google Scholar 

  12. Iwaniec, H., Kowalski, E.: Analytic number theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)

    MATH  Google Scholar 

  13. Marsaglia, G.: The structure of linear congruential sequences. In: Zaremba, S.K. (ed.) Applications of Number Theory to Numerical Analysis, pp. 249–285. Academic Press, New York (1972)

    Google Scholar 

  14. Mauduit, C., Niederreiter, H., Sárközy, A.: On pseudorandom (0,1) and binary sequences. Publ. Math. Debrecen 71(3-4), 305–324 (2007)

    MATH  MathSciNet  Google Scholar 

  15. Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences I: measures of pseudorandomness, the Legendre symbol. Acta Arith. 82, 365–377 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Moreno, C.J., Moreno, O.: Exponential sums and Goppa codes. I. Proc. Amer. Math. Soc. 111, 523–531 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)

    MATH  Google Scholar 

  18. Niederreiter, H., Rivat, J.: On the correlation of pseudorandom numbers generated by inversive methods. Monatshefte für Mathematik 153(3), 251–264 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Niederreiter, H., Winterhof, A.: Lattice structure and linear complexity of nonlinear pseudorandom numbers. Appl. Algebra Engrg. Comm. Comput. 13, 319–326 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Niederreiter, H., Winterhof, A.: On the structure of inversive pseudorandom number generators. In: Boztaş, S., Lu, H.-F. (eds.) AAECC 2007. LNCS, vol. 4851, pp. 208–216. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  21. Ostafe, A., Shparlinski, I.E.: Pseudorandomness and dynamics of Fermat quotients, 1–26 (preprint) (2009)

    Google Scholar 

  22. Wang, L.-P., Niederreiter, H.: Successive minima profile, lattice profile, and joint linear complexity profile of pseudorandom multisequences. J. Complexity 24, 144–153 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Putian University, Fujian, 351100, China

    Zhixiong Chen

  2. Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057, Zürich, Switzerland

    Alina Ostafe

  3. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, A-4040, Linz, Austria

    Arne Winterhof

Authors
  1. Zhixiong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alina Ostafe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arne Winterhof
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    M. Anwar Hasan

  2. Department of Informatics, HIB, University of Bergen, PB 7803, 5020, Bergen, Norway

    Tor Helleseth

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chen, Z., Ostafe, A., Winterhof, A. (2010). Structure of Pseudorandom Numbers Derived from Fermat Quotients. In: Hasan, M.A., Helleseth, T. (eds) Arithmetic of Finite Fields. WAIFI 2010. Lecture Notes in Computer Science, vol 6087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13797-6_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-13797-6_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13796-9

  • Online ISBN: 978-3-642-13797-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation