A method for constructing permutations, involutions and orthomorphisms with strong cryptographic properties

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.