# zbMATH — the first resource for mathematics

On the classification of APN functions up to dimension five. (English) Zbl 1184.94227
Summary: 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.

##### MSC:
 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
OEIS
Full Text:
##### 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). · Zbl 1038.94521 [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). [3] Budaghyan L., Carlet C., Leander G.: A class of quadratic apn binomials inequivalent to power functions. Cryptology ePrint Archive, Report 2006/445 (2006). · Zbl 1177.94135 [4] Budaghyan L., Carlet C., Pott A. (2006). New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inform. Theory 52: 1141–1152 · Zbl 1177.94136 · doi:10.1109/TIT.2005.864481 [5] Carlet C., Charpin P., Zinoviev V. (1998). Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15: 125–156 · Zbl 0938.94011 · doi:10.1023/A:1008344232130 [6] Dillon J.F.: APN polynomials and related codes. Banff International Research Station workshop on Polynomials over Finite Fields and Applications (2006). [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: 744–747 · Zbl 1246.11185 · doi:10.1109/TIT.2005.862128 [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). [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). [10] Knuth D.E. (1975). Estimating the efficiency of backtrack programs. Math. Comput. 29: 121–136 · Zbl 0297.68037 · doi:10.2307/2005469 [11] http://magma.maths.usyd.edu.au/magma/ (2007). [12] Nyberg K.: Differentially uniform mappings for cryptography. In: EUROCRYPT ’93, pp. 55–64 (1994). · Zbl 0951.94510 [13] Read R.C. (1978). Every one a winner. Ann. Discrete Math. 2: 107–120 · Zbl 0392.05001 · doi:10.1016/S0167-5060(08)70325-X [14] Sloane N.J.A.: The on-line encyclopedia of integer sequences. http://www.research.att.com/$$\sim$$njas/sequences/ (2007). · Zbl 1159.11327
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.