Geodesic
A method for constructing permutations, involutions and orthomorphisms with strong cryptographic properties
Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 145-151

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S-Boxes are crucial components in the design of many symmetric ciphers. To construct permutations having strong cryptographic properties is not a trivial task. In this work, we propose a new scheme based on the well-known Lai-Massey structure for generating permutations of dimension $n=2k$, $k\geq2$. The main cores of our constructions are: the inversion in $\mathrm{GF}(2^k)$, an arbitrary $k$-bit non-bijective function (which has no pre-image for $0$) and any $k$-bit permutation. Combining these components with the finite field multiplication, we provide new $8$-bit permutations without fixed points possessing a very good combination for nonlinearity, differential uniformity and minimum degree — $(104; 6; 7)$ which can be described by a system of polynomial equations with degree $3$. Also, we show that our approach can be used for constructing involutions and orthomorphisms with strong cryptographic properties.
Keywords: S-Box, Boolean functions.
Mots-clés : permutation 
@article{PDMA_2019_12_a41,
     author = {R. A. de la Cruz Jim\'enez},
     title = {A method for constructing permutations, involutions and orthomorphisms with strong cryptographic properties},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {145--151},
     publisher = {mathdoc},
     number = {12},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2019_12_a41/}
}
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R. A. de la Cruz Jiménez. A method for constructing permutations, involutions and orthomorphisms with strong cryptographic properties. Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 145-151. http://geodesic.mathdoc.fr/item/PDMA_2019_12_a41/