Binary cyclotomic generators

  • Cunsheng Ding
Session 1: Stream Ciphers-Design
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1008)


In this paper a number of binary cyclotomic generators based on cyclotomy are described. A number of cryptographic properties of the generators are controlled. A general approach to control the linear complexity and its stability for periodic sequences over any field is shown. Two bridges between number theory and stream ciphers have been established, and the relations between the design and analysis of some stream ciphers and some number-theoretic problems are shown. A number of cryptographic ideas are pointed out.


Linear Complexity Stream Cipher Output Sequence Primitive Root Basic Theorem 
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.
    T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.Google Scholar
  2. 2.
    L. D. Baumert and H. Fredricksen, The Cyclotomic Number of Order Eighteen with Applications to Difference Sets, Math. Comp. 21, 1967, pp. 204–219.Google Scholar
  3. 3.
    L. D. Baumert, Cyclic Difference Sets, Lecture Notes in Mathematics, vol. 182, Springer-Verlag, 1971.Google Scholar
  4. 4.
    T. W. Cusick, Properties of the X 2 mod N generator, to appear in IEEE Trans. Inform. Theory, 1995.Google Scholar
  5. 5.
    D. A. Cox, Primes of the Form x 2+ny 2: Fermat, Class Field Theory, and Complex Multiplication, John Wiley & Sons, 1989.Google Scholar
  6. 6.
    I. Damgård, On the Randomness of Legendre and Jacobi Sequences, Advances in Cryptology: Crypto'88, S. Goldwasser (Ed.), LNCS 403, Springer-Verlag, 1990, pp. 163–172.Google Scholar
  7. 7.
    J.-M. Deshouillers, Waring's Problem and the Circle-Method, in Number Theory and Applications, R. A. Mollin Eds., Kluwer Academic Publishers, 1989, pp. 37–44.Google Scholar
  8. 8.
    L. E. Dickson, Cyclotomy, Higher Congruences, and Waring's Problem, Amer. J. Math. 57, 1935, pp. 391–424, pp. 463–474.Google Scholar
  9. 9.
    L. E. Dickson, Solution of Waring's Problem, Amer. J. Math. 58, 1936, pp. 530–535.Google Scholar
  10. 10.
    C. Ding, G. Xiao, and W. Shan, The Stability Theory of Stream Ciphers, LNCS 561, Springer-Verlag, 1991.Google Scholar
  11. 11.
    C. Ding, The Differential Cryptanalysis and Design of the Natural Stream Ciphers, Fast Software Encryption: Proc. of the 1993 Cambridge Security Workshop, R. Anderson (Ed.), LNCS 809, Springer-Verlag, 1994, pp. 101–115.Google Scholar
  12. 12.
    E. Lehmer, On the Number of Solutions of u k+D=w2 mod p, Pacific J. Math. 5, 1955, pp. 103–118.Google Scholar
  13. 13.
    R. Lidl, H. Niederreiter, Finite Fields, in Encyclopedia of Mathematics and Its Applications, vol. 20, Addison-Wesley, 1983.Google Scholar
  14. 14.
    J. L. Massey, Shift-Register Synthesis and BCH Decoding, IEEE Trans. Inform. Theory, vol. IT-15, January, 1969, pp. 122–127.Google Scholar
  15. 15.
    W. Meier, O. Staffelbach, Nonlinearity Criteria for Cryptographic Functions, LNCS 434, Advances in Cryptology, Springer-Verlag, 1990, pp. 549–562.Google Scholar
  16. 16.
    D. Pei, Personal communications, Jan. 1994.Google Scholar
  17. 17.
    S. Pillai. On Waring's Problem, I. Ind. Math. Soc. 2, 1933, pp. 16–44.Google Scholar
  18. 18.
    W. M. Schmidt, Equations over Finite Fields: An Elementary Approach, Lecture Notes in Mathematics, vol. 536, Springer-Verlag, 1976.Google Scholar
  19. 19.
    T. Storer, Cyclotomy and Difference Sets, Marham, Chicago, 1967.Google Scholar
  20. 20.
    A. Weil, Sur les Courbes Algébriques et les Variétés qui s'en Déduisent, Actualités Sci. Ind. No. 1041.Google Scholar
  21. 21.
    A. L. Whiteman, A Family of Difference Sets, Illinois J. Math. 6, 1962, pp. 107–121.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Cunsheng Ding
    • 1
  1. 1.Department of MathematicsUniversity of TurkuTurkuFinland

Personalised recommendations