A Theory of Highly Nonlinear Functions

Horadam, K. J.

doi:10.1007/11617983_8

Cited by 4 publications

(12 citation statements)

References 14 publications

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“…This definition is used in [12][13][14] in a much wider context, and is consistent with the usual definition (see [18,10] and [5, p. 1143]) provided we switch first and second components. This can be done with no loss of generality.…”

Section: Rdss Graphs Transversals and Their Equivalencesmentioning

confidence: 71%

Relative difference sets, graphs and inequivalence of functions between groups

Horadam

2010

J of Combinatorial Designs

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For cryptographic purposes, we want to find functions with both low differential uniformity and dissimilarity to all linear functions and to know when such functions are essentially different. For vectorial Boolean functions, extended affine equivalence and the coarser Carlet-Charpin-Zinoviev (CCZ) equivalence are both used to distinguish between nonlinear functions. It remains hard to tell when CCZ equivalent functions are EA-inequivalent. This paper presents a framework for solving this problem in full generality, for functions between arbitrary finite groups. This common framework is based on relative difference sets (RDSs). The CCZ and EA equivalence classes of perfect nonlinear (PN) functions are each derived, by quite different processes, from equivalence classes of splitting semiregular RDSs. By generalizing these processes, we obtain a much strengthened formula for all the graph equivalences which define the EA equivalence class of a given function, amongst those which define its CCZ equivalence class. q

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Section: Rdss Graphs Transversals and Their Equivalencesmentioning

confidence: 71%

Relative difference sets, graphs and inequivalence of functions between groups

Horadam

2010

J of Combinatorial Designs

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“…This relationship is quite subtle, and its relationship with CCZ-equivalence has previously been stated incorrectly by the first author [17,18,20]. The mistake made in these previous publications occurred because proof depended on a factor pair (ψ, ε) defined from the transversal T φ = {(φ(x), x) : x ∈ G} of N × {1} in N × G. The possibility of (ψ, ε) also being the factor pair defined from other transversals in N × G was overlooked.…”

Section: Resultsmentioning

confidence: 99%

Bundles, presemifields and nonlinear functions

Horadam

Farmer

2008

Des. Codes Cryptogr.

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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(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 G F( p n ) 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 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 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).

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“…In [33] I initiated a study of splitting factor pairs. In the case of most interest to us, a splitting factor pair has the form (∂φ, φ) for some φ in the set C 1 (G, N) …”

Section: Bundles In the Splitting Casementioning

confidence: 99%

“…Correction to [34, 8.2, 9.2.2] A minor error was made in [33] in the fundamental definition of the bundle of a function, which has been repeated in [34,37]. The error matters only if N is nonabelian.…”

Section: Bundles In the Splitting Casementioning

confidence: 99%