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.
References
Biryukov A., Cannière C.D., Braeken A., Preneel B.: A toolbox for cryptanalysis: linear and affine equivalence algorithms. In: EUROCRYPT, pp. 33–50 (2003).
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).
Budaghyan L., Carlet C., Leander G.: A class of quadratic apn binomials inequivalent to power functions. Cryptology ePrint Archive, Report 2006/445 (2006).
Budaghyan L., Carlet C., Pott A. (2006). New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inform. Theory 52: 1141–1152
Carlet C., Charpin P., Zinoviev V. (1998). Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15: 125–156
Dillon J.F.: APN polynomials and related codes. Banff International Research Station workshop on Polynomials over Finite Fields and Applications (2006).
Edel Y., Kyureghyan G., Pott A. (2006). A new APN function which is not equivalent to a power mapping. IEEE Trans. Inform. Theory 52: 744–747
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).
dong Hou X.: Affinity of permutations of \({\mathbb{F}}_2^n\) . In: Proceedings of the Workshop on Coding and Cryptography, pp. 273–280 (2003).
Knuth D.E. (1975). Estimating the efficiency of backtrack programs. Math. Comput. 29: 121–136
Nyberg K.: Differentially uniform mappings for cryptography. In: EUROCRYPT ’93, pp. 55–64 (1994).
Read R.C. (1978). Every one a winner. Ann. Discrete Math. 2: 107–120
Sloane N.J.A.: The on-line encyclopedia of integer sequences. http://www.research.att.com/~njas/sequences/ (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of G. Leander sponsored by a DAAD postdoctoral fellowship.
Rights and permissions
About this article
Cite this article
Brinkmann, M., Leander, G. On the classification of APN functions up to dimension five. Des. Codes Cryptogr. 49, 273–288 (2008). https://doi.org/10.1007/s10623-008-9194-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-008-9194-6