zbMATH — the first resource for mathematics

A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. (English) Zbl 1194.94182
Power functions on finite fields \(GF(2^{n})\) that permute the fields have wide cryptographic applications. H. Dobbertin [Appl. Algebra Eng. Commun. Comput. 9, No. 2, 139–152 (1998; Zbl 0924.94026)] gives a list of such mappings that meet the conjectured upper bound nonlinearity which in the case of even \(n\) is \(2^{n-1}-2^{n/2}\). One of the mappings on the list is \(f(x)=x^{2^{2k}+2^k+1}\) defined on \(GF(2^{4k})\) with \(k\) odd. The authors show that \(f(x)\) has differential uniformity of \(4\) (there is no need to assume that \(k\) is odd here; if \(k\) is even the function is not a permutation). They also give a slightly different proof (from that given by Dobbertin) of the fact that \(\text{NL}(f)=2^{n-1}-2^{n/2}\) (again, the proof covers also the case of even \(k\)).

94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
14G50 Applications to coding theory and cryptography of arithmetic geometry
Full Text: DOI arXiv
[1] Dobbertin, H., One-to-one highly nonlinear power functions on \(\operatorname{GF}(2^n)\), Appl. algebra engrg. comm. comput., 9, 2, 139-152, (1998) · Zbl 0924.94026
[2] Nyberg, K., Differentially uniform mappings for cryptography, (), 55-64 · Zbl 0951.94510
[3] Matsui, M., Linear cryptanalysis method for DES cipher, (), 386-397 · Zbl 0951.94519
[4] Bracken, C.; Byrne, E.; Markin, N.; McGuire, G., New families of quadratic almost perfect nonlinear trinomials and multinomials, Finite fields appl., 14, 3, 703-714, (2008) · Zbl 1153.11058
[5] C. Bracken, E. Byrne, N. Markin, G. McGuire, A few more quadratic APN functions, Cryptography Communications, in press · Zbl 1282.11162
[6] Budaghyan, L.; Carlet, C.; Leander, G., Two classes of quadratic APN binomials inequivalent to power functions, IEEE trans. inform. theory, 54, 9, 4218-4229, (2008) · Zbl 1177.94135
[7] Budaghyan, L.; Carlet, C.; Leander, G., Constructing new APN functions from known ones, Finite fields appl., 15, 2, 150-159, (2009) · Zbl 1184.94228
[8] J.F. Dillon, APN polynomials: An update, in: 9th International Conference on Finite Fields and Applications Fq9, Dublin, Ireland, 2009
[9] C. Carlet, Vectorial (Multi-output) Boolean Functions for Cryptography, Cambridge University Press, in press · Zbl 1209.94036
[10] Canteaut, A.; Charpin, P.; Kyureghyan, G., A new class of monomial bent functions, Finite fields appl., 14, 1, 221-241, (2008) · Zbl 1162.94004
[11] Charpin, P.; Kyureghyan, G.M.M., Cubic monomial bent functions: A subclass of M, SIAM J. discrete math., 22, 2, 650-665, (2008) · Zbl 1171.11062
[12] Gold, R., Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE trans. inform. theory, 14, 1, 154-156, (1968) · Zbl 0228.62040
[13] Kasami, T., The weight enumerators for several classes of subcodes of the second order binary reed – muller codes, Inform. control, 18, 369-394, (1971) · Zbl 0217.58802
[14] Beth, T.; Ding, C., On almost perfect nonlinear permutations, (), 65-76 · Zbl 0951.94524
[15] Dillon, J.F.; Dobbertin, Hans, New cyclic difference sets with Singer parameters, Finite fields appl., 10, 3, 342-389, (2004) · Zbl 1043.05024
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.