Large Families of Pseudorandom Sequences of k Symbols and Their Complexity – Part II

  • R. Ahlswede
  • C. Mauduit
  • A. Sárközy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4123)


We continue the investigation of Part I, keep its terminology, and also continue the numbering of sections, equations, theorems etc.

Consequently we start here with Section 6. As mentioned in Section 4 we present now criteria for a triple (r,t,p) to be k–admissible. Then we consider the f–complexity (extended now to k–ary alphabets) \(\Gamma_k(\mathcal{F})\) of a family \(\mathcal{F}\). It serves again as a performance parameter of key spaces in cryptography. We give a lower bound for the f–complexity for a family of the type constructed in Part I. In the last sections we explain what can be said about the theoretically best families \(\mathcal{F}\) with respect to their f–complexity \(\Gamma_k(\mathcal{F})\). We begin with straightforward extensions of the results of [4] for k=2 to general k by using the same Covering Lemma as in [1].


Binary Sequence Prime Divisor Pseudorandom Sequence Irreducible Polynomial Acta Arith 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahlswede, R.: Coloring hypergraphs: A new approach to multi–user source coding, Part I. J. Combinatorics, Information and System Sciences 4(1), 76–115 (1979); Part II, J. Combinatorics, Information and System Sciences 5(3), 220–268 (1980)Google Scholar
  2. 2.
    Ahlswede, R.: On concepts of performance parameters for channels. In: Ahlswede, R., Bäumer, L., Cai, N., Aydinian, H., Blinovsky, V., Deppe, C., Mashurian, H. (eds.) General Theory of Information Transfer and Combinatorics. LNCS, vol. 4123, pp. 639–663. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Ahlswede, R., Winter, A.: Strong converse for identification via quantum channels. IEEE Trans. on Inform. 48(3), 569–579 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ahlswede, R., Khachatrian, L.H., Mauduit, C., Sárközy, A.: A complexity measure for families of binary sequences. Periodica Math. Hungar. 46(2), 107–118 (2003)MATHCrossRefGoogle Scholar
  5. 5.
    Cassaigne, J., Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences VII: The measures of pseudorandomness. Acta Arith. 103, 97–118 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Goubin, L., Mauduit, C., Sárközy, A.: Construction of large families of pseudo–random binary sequences. J. Number Theory 106, 56–69 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Halberstam, H., Richert, H.-E.: Sieve Methods. Academic Press, London (1974)MATHGoogle Scholar
  8. 8.
    Heath–Brown, D.R.: Artin’s conjecture for primitive roots. Quat. J. Math. 37, 27–38 (1986)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hooley, C.: On Artin’s conjecture. J. reine angew. Math. 225, 209–220 (1967)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kohayakawa, Y., Mauduit, C., Moreira, C.G., Rödl, V.: Measures of pseudorandomness for random sequences. In: Proceedings of WORDS 2003, pp. 159–169. TUCS Gen. Publ., 27, Turku Cent. Comput. Sci, Turku (2003)Google Scholar
  11. 11.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications, revised edn. Cambridge University Press, Cambridge (1994)Google Scholar
  12. 12.
    Mauduit, C., Rivat, J., Sárközy, A.: Construction of pseudorandom binary sequences using additive characters. Monatshefte Math. 141, 197–208 (2004)MATHCrossRefGoogle Scholar
  13. 13.
    Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences, I. Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82, 365–377 (1997)MATHMathSciNetGoogle Scholar
  14. 14.
    Mauduit, C., Sárközy, A.: On finite pseudorandom sequences of k symbols. Indag. Math. 13, 89–101 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Schinzel, A.: Remarks on the paper Sur certaines hypothèses concernant les nombres premiers. Acta Arith. 7, 1–8 (1961/1962)Google Scholar
  16. 16.
    Schinzel, A., Sierpiński, W.: Sur certaines hypothèses concernant les nombres premiers. Ibid 4, 185–208 (1958); Corrigendum Ibid 5, 259 (1959)Google Scholar
  17. 17.
    Weil, A.: Sur les courbes algébriques et les variétés qui s’en déduisent. Act. Sci. Ind. 1041, Hermann, Paris (1948)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • R. Ahlswede
    • 1
  • C. Mauduit
    • 2
  • A. Sárközy
    • 2
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2. 

Personalised recommendations