The One-Round Functions of the DES Generate the Alternating Group

  • Ralph Wernsdorf
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 658)


In each of the 16 DES rounds we have a permutation of 64-bitblocks. According to the corresponding key-block there are 248 possible permutations per round. In this paper we will prove that these permutations generate the alternating group. The main parts of the paper are the proof that the generated group is 3-transitive, and the application of a result from P. J. Cameron based on the classification of finite simple groups. A corollary concerning n-round functions generalizes the result.


  1. /Cam 81/.
    Cameron, P. J.: “Finite Permutation Groups and Finite Simple Groups” Bull. London Math. Soc., 13, 1981, 1–22MATHCrossRefMathSciNetGoogle Scholar
  2. /CC 91/.
    Cameron, P. J.: Cannon, J.: “Fast Recognition of Doubly Transitive Groups” Journal of Symbolic Computation, 12, Nr. 4&5, 1991, 459–474CrossRefMATHMathSciNetGoogle Scholar
  3. /CE 86/.
    Chaum, D.; Evertse, J. H.: “Cryptanalysis of DES with a reduced number of rounds; Sequences of linear factors in blockciphers” Proc. CRYPTO’ 85, Lect. Notes Comp. Sci., 218, 1986, 192–211Google Scholar
  4. /CG 75/.
    Coppersmith, D.; Grossman, E.: “Generators for certain alternating groups with applications to cryptography” Journal of Applied Mathematics, 29, Nr. 4, 1975, 624–627MATHMathSciNetGoogle Scholar
  5. /DDF 84/.
    Davio, M.; Desmedt, Y.: Fosseprez, M. et al.: “Analytical Characteristics of the DES” Proc. CRYPTO’ 83, Plenum Press, New York and London, 1984, 171–202Google Scholar
  6. /EG 83/.
    Even, S.; Goldreich, O.: “DES-like functions can generate the alternating group” IEEE Transactions on Information Theory, IT-29, Nr. 6, 1983, 863–865CrossRefMathSciNetGoogle Scholar
  7. /KRS 88/.
    Kaliski, B. S.; Rivest, R. L.; Sherman, A. T.: “Is the Data Encryption Standard a Group? (Results of Cycling Experiments on DES)” Journal of Cryptology, 1, Nr. 1, 1988, 3–36MATHCrossRefMathSciNetGoogle Scholar
  8. /NBS 77/.
    National Bureau of Standards: “Data Encryption Standard” FIPS PUB 46, Washington, 1977Google Scholar
  9. /PZ 90/.
    Pieprzyk, J.; Zhang, X. M.: “Permutation Generators of Alternating Groups” Proc. AUSCRYPT’ 90, Lect. Notes Comp. Sci., 453, 1990, 237–244Google Scholar
  10. /RM 85/.
    Reeds, J. A.; Manferdelli, J. L.: “DES has no per round linear factors” Proc. CRYPTO’ 84, Lect. Notes Comp. Sci., 196, 1985, 377–389Google Scholar
  11. /Rob 82/.
    Robinson, D. J. S.: “A Course in the Theory of Groups” Springer, New York, Heidelberg, Berlin, 1982MATHGoogle Scholar
  12. /SM 87/.
    Simmons, G. J.; Moore, J. H.: “Cycle structure of the DES for keys having palindromic (or antipalindromic) sequences of round keys” IEEE Transactions on Software Engineering, SE-13, Nr. 2, 1987, 262–273CrossRefGoogle Scholar
  13. /Wie 64/.
    Wielandt, H.: “Finite Permutation Groups” Academic Press, New York, London, 1964MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Ralph Wernsdorf
    • 1
  1. 1.SIT Gesellschaft für Systeme der Informationstechnik mbHGrünheide (Mark)Germany

Personalised recommendations