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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
Magma
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