Bundles, presemifields and nonlinear functions

Abstract

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 GF(2 n ), affine bundles coincide with EA-equivalence classes. From equivalence classes ("bundles") of presemifields of order p n , we derive bundles of functions over GF(p n ) of the form ?(x)*?(x), where ?, ? are linearised permutation polynomials and * is a presemifield multiplication. We prove there are exactly p bundles of presemifields of order p 2 and give a representative of each. We compute all bundles of presemifields of orders p n ? 27 and in the isotopism class of GF(32) and we measure the differential uniformity of the derived ?(x)*?(x). This technique produces functions with low differential uniformity, including PN functions (p odd), and quadratic APN and differentially 4-uniform functions (p = 2).