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On known and new differentially uniform functions. (English) Zbl 1279.94060
Parampalli, Udaya (ed.) et al., Information security and privacy. 16th Australasian conference, ACISP 2011, Melbourne, Australia, July 11–13, 2011. Proceedings. Berlin: Springer (ISBN 978-3-642-22496-6/pbk). Lecture Notes in Computer Science 6812, 1-15 (2011).
Summary: We give a survey on the constructions of APN and differentially 4-uniform functions suitable for designing S-boxes for block ciphers. We recall why the search for more of such functions is necessary. We propose a way of designing functions which can possibly be APN or differentially 4-uniform and be bijective. We illustrate it with an example of a differentially 4-uniform $$(n,n)$$-permutation for $$n$$ odd, based on the power function $$x ^{3}$$ over the second order Galois extension of $${\mathbb F}_{2^{n+1}}$$, and related to the Dickson polynomial $$D _{3}$$ over this field. These permutations have optimal algebraic degree and their nonlinearity happens to be rather good (but worse than that of the multiplicative inverse functions).
For the entire collection see [Zbl 1217.94003].

##### MSC:
 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
##### Keywords:
block cipher; vectorial Boolean function; S-box
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