Quasigroup Representation of Some Feistel and Generalized Feistel Ciphers

  • Aleksandra MilevaEmail author
  • Smile Markovski
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 207)


There are several block ciphers designed by using Feistel networks or their generalization, and some of them allow to be represented by using quasigroup transformations, for suitably defined quasigroups. We are interested in those Feistel ciphers and Generalized Feistel ciphers whose round functions in their Feistel networks are bijections. In that case we can define the wanted quasigroups by using suitable orthomorphisms, derived from the corresponding Feistel networks. Quasigroup representations of the block ciphers MISTY1, Camellia, Four-Cell +  and SMS4 are given as examples.


Feistel network Feistel cipher orthomorphism quasigroup MISTY1 Camellia Four-Cell +  SMS4 


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  1. 1.
    Aoki, K., Ichikawa, T., Kanda, M., Matsui, M., Moriai, S., Nakajima, J., Tokita, T.: Camellia: A 128-bit block cipher suitable for multiple platforms - design and analysis. In: Stinson, D.R., Tavares, S. (eds.) SAC 2000. LNCS, vol. 2012, pp. 39–56. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Adams, C.M., Tavares, S.E.: Designing S-boxes for Ciphers Resistant to Differential Cryptanalysis. In: 3rd Symposium on State and Progress of Research in Cryptography, Rome, Italy, pp. 181–190 (1993)Google Scholar
  3. 3.
    Brown, L., Kwan, M., Pieprzyk, J., Seberry, J.: Improving Resistance to Differential Cryptanalysis and the Redesign of LOKI. In: Imai, H., Rivest, R.L., Matsumoto, T. (eds.) ASIACRYPT 1991. LNCS, vol. 739, pp. 36–50. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  4. 4.
    Choy, J., Chew, G., Khoo, K., Yap, H.: Cryptographic Properties and Application of a Generalized Unbalanced Feistel Network Structure. In: Boyd, C., González Nieto, J. (eds.) ACISP 2009. LNCS, vol. 5594, pp. 73–89. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Choy, J., Chew, G., Khoo, K., Yap, H.: Cryptographic Properties and Applications of a Generalized Unbalnced Feistel Network Structure. Cryptography and Communications 3(3), 141–164 (2011) (revised version)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
  7. 7.
    Gligoroski, D., Andova, S., Knapskog, S.J.: On the Importance of the Key Separation Principle for Different Modes of Operation. In: Chen, L., Mu, Y., Susilo, W. (eds.) ISPEC 2008. LNCS, vol. 4991, pp. 404–418. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Denes, J., Keedwell, A.D.: Latin squares: New developments in the theory and applications. Elsevier science publishers (1991)Google Scholar
  9. 9.
    Diffie, W., Ledin, G. (trans.): SMS4 encryption algorithm for wireless networks. Cryptology ePrint Archive, Report 2008/329 (2008)Google Scholar
  10. 10.
    Evans, A.B.: Orthomorphism Graphs of Groups. Journal of Geometry 35(1-2), 66–74 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Feistel, H.: Cryptography and computer privacy. Scientific American 228(5), 15–23 (1973)CrossRefGoogle Scholar
  12. 12.
    International Standard - ISO/IEC 18033-3, Information technology - Security techniques - Encryption algorithms - Part 3: Block ciphers (2005)Google Scholar
  13. 13.
    Kwan, M.: The Design of the ICE Encryption Algorithm. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 69–82. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  14. 14.
    NBS FIPS PUB 46: Data Encryption Standard. National Bureau of Standards, U.S. Department of Commerce (1977)Google Scholar
  15. 15.
    NESSIE-New European Schemes for Signatures, Integrity, and Encryption, final report of European project IST-1999-12324, Version 0.15, Archive (April 19, 2004),
  16. 16.
    Markovski, S., Gligoroski, D., Andova, S.: Using quasigroups for one-one secure encoding. In: VIII Conf. Logic and Computer Science, LIRA 1997, Novi Sad, Serbia, pp. 157–162 (1997)Google Scholar
  17. 17.
    Matsui, M.: New block encryption algorithm MISTY. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 54–68. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  18. 18.
    Merkle, R.C.: Fast Software Encryption Functions. In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 476–500. Springer, Heidelberg (1991)Google Scholar
  19. 19.
    Mihajloska, H., Gligoroski, D.: Construction of Optimal 4-bit S-boxes by Quasigroups of Order 4. In: SECURWARE 2012, Rome, Italy (2012)Google Scholar
  20. 20.
    Mileva, A., Markovski, S.: Shapeless quasigroups derived by Feistel orthomorphisms. Glasnik Matematicki (accepted for printing) Google Scholar
  21. 21.
    Sade, A.: Quasigroups automorphes par le groupe cyclique. Canadian Journal of Mathematics 9, 321–335 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Schneier, B.: Description of a New Variable-Length Key, 64-Bit Block Cipher (Blowfish). In: Fast Software Encryption, Cambridge Security Workshop Proceedings, pp. 191–204. Springer (1994)Google Scholar
  23. 23.
    Zheng, Y., Matsumoto, T., Imai, H.: On the construction of block ciphers provably secure and not relying on any unproved hypotheses. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 461–480. Springer, Heidelberg (1990)Google Scholar

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Authors and Affiliations

  1. 1.Faculty of Computer ScienceUniversity “Goce Delčev”ŠtipRepublic of Macedonia
  2. 2.Faculty of Computer Science and EngineeringUniversity “Ss Cyril and Methodius”SkopjeRepublic of Macedonia

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