A new tool for assurance of perfect nonlinearity. (English) Zbl 1206.94048

Golomb, Solomon W. (ed.) et al., Sequences and their applications – SETA 2008. 5th international conference, Lexington, KY, USA, September 14–18, 2008 Proceedings. Berlin: Springer (ISBN 978-3-540-85911-6/pbk). Lecture Notes in Computer Science 5203, 415-419 (2008).
Summary: Let \(f(x)\) be a mapping \(f: \text{GF}(p ^{n }) \rightarrow \text{GF}(p ^{n })\), where \(p\) is prime and \(\text{GF}(p ^{n })\) is the finite field with \(p ^{n }\) elements. A mapping \(f\) is called differentially \(k\)-uniform if \(k\) is the maximum number of solutions \(x \in \text{GF}(p ^{n })\) of \(f(x + a) - f(x) = b\), where \(a, b \in \text{GF}(p ^{n })\) and \(a \neq 0\). A 1-uniform mapping is called perfect nonlinear (PN). In this paper, we propose an approach for assurance of perfect nonlinearity which involves simply checking a trace condition.
For the entire collection see [Zbl 1155.94003].


94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
Full Text: DOI


[1] Carlet, C.; Ding, C., Highly nonlinear mappings, J. of Complexity, 20, 205-244 (2004) · Zbl 1053.94011 · doi:10.1016/j.jco.2003.08.008
[2] Coulter, R. S.; Matthews, R. W., Planar functions and planes of Lenz-Barlotti class II, Des., Codes, Cryptogr., 10, 167-184 (1997) · Zbl 0872.51007 · doi:10.1023/A:1008292303803
[3] Coulter, R. S.; Henderson, M., Commutative presemifields and semifields, Advances in Math., 217, 282-304 (2008) · Zbl 1194.12007 · doi:10.1016/j.aim.2007.07.007
[4] Dembowski, P.; Ostrom, T., Planes of order n with collineation groups of order n^2, Math. Z., 103, 239-258 (1968) · Zbl 0163.42402 · doi:10.1007/BF01111042
[5] Ding, C.; Yuan, J., A family of skew Paley-Hadamard difference sets, J. Comb. Theory Ser. A, 113, 1526-1535 (2006) · Zbl 1106.05016 · doi:10.1016/j.jcta.2005.10.006
[6] Helleseth, T.; Sandberg, D., Some power mappings with low differential uniformity, Applicable Algebra in Engineering, Communications and Computing, 8, 363-370 (1997) · Zbl 0886.11067 · doi:10.1007/s002000050073
[7] Helleseth, T.; Rong, C.; Sandberg, D., New families of almost perfect nonlinear power mappings, IEEE Trans. Inform. Theory, 52, 475-485 (1999) · Zbl 0960.11051 · doi:10.1109/18.748997
[8] Helleseth, T., Kyureghyan, G., Ness, G.J., Pott, A.: On a family of perfect nonlinear binomials (submitted) · Zbl 1198.94098
[9] Lidl, R.; Niederreiter, H., Finite Fields (1997), Cambridge: Cambridge University Press, Cambridge
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