zbMATH — the first resource for mathematics

Constructing new APN functions from known ones. (English) Zbl 1184.94228
Summary: We present a method for constructing new quadratic APN functions from known ones. Applying this method to the Gold power functions we construct an APN function \(x^3+\text{tr}(x^9)\) over \(\mathbb F_{2^n}\). It is proven that for \(n\geq 7\) this function is CCZ-inequivalent to the Gold functions, and in the case \(n=7\) it is CCZ-inequivalent to any power mapping (and, therefore, to any APN function belonging to one of the families of APN functions known so far).

94D10 Boolean functions
94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
Full Text: DOI
[1] Biham, E.; Shamir, A., Differential cryptanalysis of DES-like cryptosystems, J. cryptology, 4, 1, 3-72, (1991) · Zbl 0729.68017
[2] Bracken, C.; Byrne, E.; Markin, N.; McGuire, G., New families of quadratic almost perfect nonlinear trinomials and multinomials, Finite fields appl., 14, 703-714, (2008) · Zbl 1153.11058
[3] Bracken, C.; Byrne, E.; Markin, N.; McGuire, G., A few more quadratic APN functions, preprint, 2008. Available at
[4] K. Browning, J.F. Dillon, R.E. Kibler, M. McQuistan, APN polynomials and related codes, to appear in a special volume of J. Combin. Inform. System Sci., 2008, in press; honoring the 75th birthday of Prof. D.K. Ray-Chaudhuri · Zbl 1269.94035
[5] Budaghyan, L., The simplest method for constructing APN polynomials EA-inequivalent to power functions, (), 177-188 · Zbl 1177.94133
[6] Budaghyan, L.; Carlet, C., Classes of quadratic APN trinomials and hexanomials and related structures, IEEE trans. inform. theory, 54, 5, 2354-2357, (May 2008)
[7] Budaghyan, L.; Carlet, C.; Leander, G., Two classes of quadratic APN binomials inequivalent to power functions, IEEE trans. inform. theory, 54, 9, 4218-4229, (Sept. 2008)
[8] Budaghyan, L.; Carlet, C.; Pott, A., New classes of almost bent and almost perfect nonlinear functions, IEEE trans. inform. theory, 52, 3, 1141-1152, (March 2006)
[9] Canteaut, A.; Charpin, P.; Dobbertin, H., Weight divisibility of cyclic codes, highly nonlinear functions on \(\mathbb{F}_{2^m}\), and crosscorrelation of maximum-length sequences, SIAM J. discrete math., 13, 1, 105-138, (2000) · Zbl 1010.94013
[10] C. Carlet, Vectorial Boolean functions for cryptography, chapter of the monography Boolean Methods and Models, Y. Crama, P. Hammer (Eds.), Cambridge Univ. Press, in press
[11] Carlet, C.; Charpin, P.; Zinoviev, V., Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. codes cryptogr., 15, 2, 125-156, (1998) · Zbl 0938.94011
[12] Chabaud, F.; Vaudenay, S., Links between differential and linear cryptanalysis, (), 356-365 · Zbl 0879.94023
[13] J.F. Dillon, APN Polynomials and Related Codes, Polynomials over Finite Fields and Applications, Banff International Research Station, Nov. 2006
[14] J.F. Dillon, private communication, Feb. 2007
[15] Dobbertin, H., Almost perfect nonlinear power functions over \(\mathit{GF}(2^n)\): the niho case, Inform. and comput., 151, 57-72, (1999) · Zbl 1072.94513
[16] Dobbertin, H., Almost perfect nonlinear power functions over \(\mathit{GF}(2^n)\): a new case for n divisible by 5, (), 113-121 · Zbl 1010.94550
[17] Edel, Y.; Pott, A., A new almost perfect nonlinear function which is not quadratic, preprint, 2008. Available at
[18] Lachaud, G.; Wolfmann, J., The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE trans. inform. theory, 36, 686-692, (1990) · Zbl 0703.94011
[19] Matsui, M., Linear cryptanalysis method for DES cipher, (), 386-397 · Zbl 0951.94519
[20] Nyberg, K., Differentially uniform mappings for cryptography, (), 55-64 · Zbl 0951.94510
[21] Nyberg, K., S-boxes and round functions with controllable linearity and differential uniformity, (), 111-130 · Zbl 0939.94559
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.