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Designs, Codes and Cryptography

, Volume 49, Issue 1–3, pp 273–288 | Cite as

On the classification of APN functions up to dimension five

  • Marcus Brinkmann
  • Gregor Leander
Article

Abstract

We classify the almost perfect nonlinear (APN) functions in dimensions 4 and 5 up to affine and CCZ equivalence using backtrack programming and give a partial model for the complexity of such a search. In particular, we demonstrate that up to dimension 5 any APN function is CCZ equivalent to a power function, while it is well known that in dimensions 4 and 5 there exist APN functions which are not extended affine (EA) equivalent to any power function. We further calculate the total number of APN functions up to dimension 5 and present a new CCZ equivalence class of APN functions in dimension 6.

Keywords

APN functions Backtrack Affine equivalence CCZ equivalence Affine subspaces 

AMS Classification

11T71 

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References

  1. Biryukov A., Cannière C.D., Braeken A., Preneel B.: A toolbox for cryptanalysis: linear and affine equivalence algorithms. In: EUROCRYPT, pp. 33–50 (2003).Google Scholar
  2. Budaghyan L., Carlet C., Felke P., Leander G.: An infinite class of quadratic APN functions which are not equivalent to power mappings. In: IEEE International Symposium on Information Theory, pp. 2637–2641 (2006).Google Scholar
  3. Budaghyan L., Carlet C., Leander G.: A class of quadratic apn binomials inequivalent to power functions. Cryptology ePrint Archive, Report 2006/445 (2006).Google Scholar
  4. Budaghyan L., Carlet C., Pott A. (2006). New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inform. Theory 52: 1141–1152CrossRefMathSciNetGoogle Scholar
  5. Carlet C., Charpin P., Zinoviev V. (1998). Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15: 125–156zbMATHCrossRefMathSciNetGoogle Scholar
  6. Dillon J.F.: APN polynomials and related codes. Banff International Research Station workshop on Polynomials over Finite Fields and Applications (2006).Google Scholar
  7. Edel Y., Kyureghyan G., Pott A. (2006). A new APN function which is not equivalent to a power mapping. IEEE Trans. Inform. Theory 52: 744–747CrossRefMathSciNetGoogle Scholar
  8. Faradžev I.A.: Constructive enumeration of combinatorial objects. In: Problèmes Combinatoires et Théorie des Graphes, vol. 260, pp. 131–135. Coloques internationaux C.N.R.S. (1978).Google Scholar
  9. dong Hou X.: Affinity of permutations of \({\mathbb{F}}_2^n\) . In: Proceedings of the Workshop on Coding and Cryptography, pp. 273–280 (2003).Google Scholar
  10. Knuth D.E. (1975). Estimating the efficiency of backtrack programs. Math. Comput. 29: 121–136zbMATHCrossRefMathSciNetGoogle Scholar
  11. Nyberg K.: Differentially uniform mappings for cryptography. In: EUROCRYPT ’93, pp. 55–64 (1994).Google Scholar
  12. Read R.C. (1978). Every one a winner. Ann. Discrete Math. 2: 107–120zbMATHCrossRefMathSciNetGoogle Scholar
  13. Sloane N.J.A.: The on-line encyclopedia of integer sequences. http://www.research.att.com/~njas/sequences/ (2007).

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Ruhr-Universität BochumBochumGermany
  2. 2.University of ToulonLa GardeFrance

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