an:05536765
Zbl 1178.94190
Horadam, K. J.; Farmer, D. G.
Bundles, presemifields and nonlinear functions
EN
Des. Codes Cryptography 49, No. 1-3, 79-94 (2008).
00247954
2008
j
94A60 12K10 11T71 20J06
Summary: 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 \(\text{GF}(2^n)\), affine bundles coincide with EA-equivalence classes. From equivalence classes (``bundles'') of presemifields of order \(p^n\), we derive bundles of functions over \(\text{GF}(p^n)\) of the form \(\lambda(x)*\rho(x)\), where \(\lambda, \rho\) 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 \(p^n\leq 27\) and in the isotopism class of \(\text{GF}(32)\) and we measure the differential uniformity of the derived \(\lambda(x)*\rho(x)\). This technique produces functions with low differential uniformity, including PN functions (\(p\) odd), and quadratic APN and differentially 4-uniform functions \((p=2)\).