Bundles, presemifields and nonlinear functions

Bundles are equivalence classes of functions derived from equivalence classes of transversals. They preserve measures of resistance to differential and linear cryptanalysis. For functions over G F(2n), affine bundles coincide with EA-equivalence classes. From equivalence classes (“bundles”) of presemifields of order pn , we derive bundles of functions over G F(pn) of the form λ(x)∗ρ(x), where λ, ρ are linearised permutation polynomials and ∗ is a presemifield multiplication. We prove there are exactly p bundles of presemifields of order p2 and give a representative of each. We compute all bundles of presemifields of orders pn ≤ 27 and in the isotopism class of G F(32) and we measure the differential uniformity of the derived λ(x) ∗ ρ(x). This technique produces functions with low differential uniformity, including PN functions (p odd), and quadratic APN and differentially 4-uniform functions (p = 2).