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Bundles, presemifields and nonlinear functions. (English) Zbl 1178.94190
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 \((p=2)\).

MSC:
94A60 Cryptography
12K10 Semifields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
20J06 Cohomology of groups
Software:
Magma
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