On 3-to-1 and Power APN S-Boxes

  • Deepak Kumar Dalai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5203)


Almost Perfect Nonlinear (APN) S-boxes are used in block ciphers to prevent differential attacks. The non-evidence of permutation APN S-box on even number of variables and the efficiency of power functions bring the importance of power APN S-boxes to use in block ciphers. We present a special class of 3-to-1 S-box (named as S3-to-1 S-box) on even number of variables. The power APN S-boxes on even number of variables fall in this class. Further, another important class of APN functions X3 + tr(X9) too falls in this class. We study some results of S3-to-1 S-boxes. In another section we present a necessary condition for power functions to be APN. Using this necessary condition we can filter out some non-APN power functions. Specifically, if the number of variables is multiple of small primes, then one can filter out many non-APN functions.


S-box Power Function APN Function Differential Cryptanalysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berger, T.P., Canteaut, A., Charpin, P., Laigle-Chapuy, Y.: Almost Perfect Nonlinear functions. IEEE Trans. Inform. Theory 52(9), 4160–4170 (2006)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Biham, E., Shamir, A.: Differential Cryptanalysis of DES-like Cryptosystem. Journal of Cryptology 4(1), 3–72 (1991)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Budaghyan, L., Carlet, C., Leander, G.: Constructing new APN functions from known ones. Cryptology ePrint Archive: report 2007/063Google Scholar
  4. 4.
    Carlet, C., Charpin, P., Zinoviev, V.: Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Charpin, P., Tietävämen, A., Zinoviev, V.: On binary cyclic codes with minimum distance d = 3. Problems Inform. Transmission 33(4), 287–296 (1997)MATHMathSciNetGoogle Scholar
  6. 6.
    Comtet, L.: Advanced combinatorics. Reidel Publication (1974)Google Scholar
  7. 7.
    Nyberg, K., Knudsen, L.R.: Provable security against differential cryptanalysis. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 566–574. Springer, Heidelberg (1993)Google Scholar
  8. 8.
    Nyberg, K.: Differentially uniform mappings for cryptography. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Deepak Kumar Dalai
    • 1
  1. 1.Applied Statistics Unit, Indian Statistical InstituteCalcuttaIndia

Personalised recommendations