Direct sums of balanced functions, perfect nonlinear functions, and orthogonal cocycles

LeBel, Alain; Horadam, K. J.

doi:10.1002/jcd.20187

Cited by 4 publications

(4 citation statements)

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“…8.3] in the negative for the case p = 2. All orthogonal cocycles on Z 4 2 are multiplicative [25] and there are 32 + 1190 + 224 = 1446 bundles of multiplicative orthogonal cocycles on Z 4 2 , but this is clearly not a power of 2. The result for G F(32) shows that even on restricting to Galois Fields, the number of bundles is not a power of 2.…”

Section: Definition 4 [21 Definition 32] Set G = (Gmentioning

confidence: 95%

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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Section: Definition 4 [21 Definition 32] Set G = (Gmentioning

confidence: 95%

Bundles, presemifields and nonlinear functions

Horadam

Farmer

2008

Des. Codes Cryptogr.

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“…(For an example when p = 3, see the remarks after Corollary 3.12.) For p = 2, LeBel [19,20] (see also [13, §9.3.1.1]) showed that all orthogonal cocycles are multiplicative for n ≤ 3, and claimed to show the same for n = 4. Dillon [7] has since informed us that he has found examples which contradict LeBel's argument for the case n = 4, but he has also established computationally that the claim is correct.…”

Section: Relative Difference Sets and Presemifieldsmentioning

confidence: 97%

“…We would very much like to thank John Dillon for discovering and correcting errors in results in [19, Chapter 7], repeated in [13, §9.3.1.1] and [20], which we had quoted in an earlier version of this paper. We also thank the referees for excellent suggestions which greatly improved the clarity of our exposition.…”

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confidence: 91%

Equivalence classes of multiplicative central (p n , p n , p n , 1)-relative difference sets

Farmer

Horadam

2010

Cryptogr. Commun.

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We show by explicit construction that the equivalence classes of multiplicative central ( p n , p n , p n , 1)-RDSs relative to Z n p in groups E with E/Z n p ∼ = Z n p are in one-to-one correspondence with the strong isotopism classes of presemifields of order p n . We also show there are 1,446 equivalence classes of central (16, 16, 16, 1)-RDS relative to Z 4 2 , in groups E for which E/Z 4 2 ∼ = Z 4 2 . Only one is abelian.

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“…If U = × r j=1 U j and ψ ∈ Z(G, U ) is orthogonal then so too is each projection ψ j ∈ Z(G, U j ). (For a converse statement see [13,Theorem 3.4].) Suppose that Z(G, U k ) contains exactly t k orthogonal cocycles; by testing a space of size t 1 · · · t r we will locate all orthogonal elements of Z(G, U ).…”

Section: Orthogonal Cocyclesmentioning

confidence: 98%