Constructing $8$-bit permutations, $8$-bit involutions and $8$-bit orthomorphisms with almost optimal cryptographic parameters

Nonlinear bijective transformations are crucial components in the design of many symmetric ciphers. To construct permutations having cryptographic properties close to the optimal ones is not a trivial problem. We propose a new construction based on the well-known Lai – Massey structure for generating binary permutations of dimension $n=2k$, $k\geq2$. The main cores of our constructions are: the inversion in $\mathbb{F}_{2^k}$, an arbitrary $k$-bit non-bijective function (which has no preimage for $0$) and any $k$-bit permutation. Combining these components with the finite field multiplication, we provide new $8$-bit permutations with high values of its basic cryptographic parameters. Also, we show that our approach may be used for constructing $8$-bit involutions and $8$-bit orthomorphisms that have strong cryptographic properties.