Advertisement

The group generated by the round functions of a GOST-like cipher

  • R. Aragona
  • A. Caranti
  • M. Sala
Article

Abstract

We define a cipher that is an extension of GOST, and study the permutation group generated by its round functions. We show that, under minimal assumptions on the components of the cipher, this group is the alternating group on the plaintext space. This we do by first showing that the group is primitive, and then applying the O’Nan-Scott classification of primitive groups.

Keywords

Cryptosystems Feistel networks GOST round functions primitive groups wreath products 

Mathematics Subject Classification

20B15 20B35 94A60 

Notes

Acknowledgments

The authors are grateful to the referee for her suggestions. The authors are indebted to Rüdiger Sparr and Ralph Wernsdorf for reading a previous version and suggesting several changes, pointing out in particular a serious oversight on our part regarding the parity of permutations and providing a shorter argument for Sect. 5.3.

References

  1. 1.
    Aragona, R., Caranti, A., Dalla Volta, F., Sala, M.: On the group generated by the round functions of translation based ciphers over arbitrary finite fields. Finite Fields Appl. 25, 293–305 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Caranti, A., Dalla Volta, F., Sala, M.: An application of the O’Nan-Scott theorem to the group generated by the round functions of an AES-like cipher. Des. Codes Cryptogr. 52(3), 293–301 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Caranti, A., Dalla Volta, F., Sala, M.: On some block ciphers and imprimitive groups. Appl. Algebra Eng. Comm. Comput. 20(5–6), 339–350 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Coppersmith, D., Grossman, Edna: Generators for certain alternating groups with applications to cryptography. SIAM J. Appl. Math. 29(4), 624–627 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dolmatov, V.: GOST 28147–89: encryption, decryption, and message authentication code (MAC) algorithms. Technical report (2010). http://tools.ietf.org/html/rfc5830
  6. 6.
    Even, S., Goldreich, O.: DES-like functions can generate the alternating group. IEEE Trans. Inform. Theory 29(6), 863–865 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Goursat, E.: Sur les substitutions orthogonales et les divisions régulières de l’espace. Ann. Sci. École Norm. Sup. 6(3), 9–102 (1889)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Guralnick, R.M.: Subgroups of prime power index in a simple group. J. Algebra 81(2), 304–311 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kaliski Jr., B.S., Rivest, R.L., Sherman, A.T.: Is the data encryption standard a group? (Results of cycling experiments on DES). J. Cryptol. 1(1), 3–36 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, C.H.: The finite primitive permutation groups containing an abelian regular subgroup. Proc. Lond. Math. Soc. 87(3), 725–747 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liebeck, M.W., Praeger, C.E., Saxl, J.: On the O’Nan-Scott theorem for finite primitive permutation groups. J. Aust. Math. Soc. Ser. A 44(3), 389–396 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Murphy, S., Paterson, K., Wild, P.: A weak cipher that generates the symmetric group. J. Cryptol. 7(1), 61–65 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Oliynykov, R.: Cryptanalysis of symmetric block ciphers based on the Feistel network with non-bijective S-boxes in the round function. Cryptology ePrint Archive, Report 2011/685 (2011). https://eprint.iacr.org/2011/685
  14. 14.
    Paterson, K.G.: Imprimitive permutation groups and trapdoors in iterated block ciphers, Fast software encryption, LNCS, vol. 1636, pp. 201–214. Springer, Berlin (1999)zbMATHGoogle Scholar
  15. 15.
    Petrillo, J.: Goursat’s other theorem. Coll. Math. J. 40(2), 119–124 (2009)CrossRefGoogle Scholar
  16. 16.
    Sparr, R., Wernsdorf, R.: Group theoretic properties of Rijndael-like ciphers. Discrete Appl. Math. 156(16), 3139–3149 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sparr, R., Wernsdorf, R.: The round functions of KASUMI generate the alternating group. J. Math. Cryptol. 9(1), 23–32 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wernsdorf, R.: The one-round functions of the DES generate the alternating group, Advances in cryptology–EUROCRYPT ’92 (Balatonfüred: Lecture Notes in Computer Science, vol. 658. Springer, Berlin (1992)Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di TrentoTrentoItaly

Personalised recommendations