Dickson Pseudoprimes and Primality Testing

  • Winfried B. Müller
  • Alan Oswald
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 547)


The paper gives a general definition for the concept of strong Dickson pseudoprimes which contains as special cases the Carmichael numbers and the strong Fibonacci pseudoprimes. Furthermore, we give necessary and sufficient conditions for two important classes of strong Dickson pseudoprimes and deduce some properties for their elements. A suggestion of how to improve a primality test by Baillie&Wagstaff concludes the paper.


Arbitrary Integer Primality Test Probabilistic Primality Negative Parameter Permutation Polynomial 
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.


  1. [1]
    Chernick J.: On Fermat’s simple theorem. Bull.Amer.Math.Soc. 45, 269–274 (1939).zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Dubner H.: A New Method for Producing Large Carmichael Numbers. Math.Comp. 53, No. 187, 411–414 (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Baillie, R., Wagstaff Jr., S.S.: Lucas pseudoprimes. Math.Comp. 35, 1391–1417 (1980).zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Di Porto, A., Filipponi, P.: A Probabilistic Primality Test Based on the Properties of Certain Generalized Lucas Numbers. In: Advances in Cryptology — Eurocrypt’88, Lecture Notes in Computer Science 330, Springer-Verlag, New York-Berlin-Heidelberg, pp. 211–223, 1988.Google Scholar
  5. [5]
    Filipponi, P.: Table of Fibonacci Pseudoprimes to 108. Note Recensioni Notizie 37, No. 1–2, 33–38 (1988).Google Scholar
  6. [6]
    Jaeschke, G.: Math.Comp. 55, No. 191, 383–389 (1990).zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Koblitz, N.: A Course in Number Theory and Cryptography. Springer-Verlag, New York-Berlin-Heidelberg, 1987.zbMATHGoogle Scholar
  8. [8]
    Lidl, R., MÜller, W.B.: Permutation polynomials in RSA-cryptosystems. Advances in Cryptology-Crypto’ 83 (ed. D. Chaum), New York, Plenum Press, pp. 293–301, 1984.Google Scholar
  9. [9]
    Lidl, R., MÜller, W.B.: Generalizations of the Fibonacci Pseudoprimes Test. To appear in Discrete Mathem. 92 (1991).Google Scholar
  10. [10]
    Lidl, R., MÜller, W.B., Oswald A.: Some Remarks on Strong Fibonacci Pseudoprimes. Applicable Algebra in Engineering, Communication and Computing (AAECC) 1, 59–65 (1990).zbMATHCrossRefGoogle Scholar
  11. [11]
    MÜller, W.B.: Polynomial Functions in Modern Cryptology. In: Contributions to General Algebra 3, Teubner-Verlag, Stuttgart, pp. 7–32, 1985.Google Scholar
  12. [12]
    MÜller, W.B., NÖbauer R.: Cryptanalysis of the Dickson-scheme. In: Advances in Cryptology — Eurocrypt’85, Lecture Notes in Computer Science 219, Springer-Verlag, New York-Berlin-Heidelberg, pp. 50–61, 1986.Google Scholar
  13. [13]
    NÖbauer, W.: Über die Fixpunkte der Dickson—Permutationen. Sb.d.Österr.Akad.-d.Wiss., math.-nat.Kl., Abt.II, Bd. 193, 115–133 (1984).zbMATHGoogle Scholar
  14. [14]
    Ribenboim, P.: The Book of Prime Number Records. Springer-Verlag, New York-Berlin-Heidelberg, 1988.zbMATHGoogle Scholar
  15. [15]
    Singmaster, D.: Some Lucas pseudoprimes. Abstracts Amer.Math.Soc. 4, No.83T-10-146, p.197 (1983).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Winfried B. Müller
    • 1
  • Alan Oswald
    • 2
  1. 1.Institut für MathematikUniversität KlagenfurtKlagenfurtAustria
  2. 2.School of Computing and MathematicsTeesside PolytechnicMiddlesbrough, ClevelandGreat Britain

Personalised recommendations