Highly nonlinear mappings

Carlet, Claude; Ding, Cunsheng

doi:10.1016/j.jco.2003.08.008

Cited by 212 publications

(158 citation statements)

References 66 publications

(89 reference statements)

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“…Functions f : G → N , for which the equation f (x + a) − f (x) = b has no more than 1 solution for all a, b ∈ G not equal to zero simultaneously, are called differentially 1-uniform [25]. If N satisfies |G|/|N | = m ∈ N and function f : G → N is such that |{x ∈ G | f (x + a) − f (x) = b}| = m for any b ∈ N and non-zero a ∈ G, the latter is called perfect non-linear [7]. These functions are used in cryptography to construct S-boxes that are not vulnerable to differential cryptanalysis.…”

Section: Related Combinatorial Structuresmentioning

confidence: 99%

A Criterion for Attaining the Welch Bounds with Applications for Mutually Unbiased Bases

Belovs

Smotrovs

2008

Mathematical Methods in Computer Science

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The paper gives a short introduction to mutually unbiased bases and the Welch bounds and demonstrates that the latter is a good technical tool to explore the former. In particular, a criterion for a system of vectors to satisfy the Welch bounds with equality is given and applied for the case of MUBs. This yields a necessary and sufficient condition on a set of orthonormal bases to form a complete system of MUBs.This condition takes an especially elegant form in the case of homogeneous systems of MUBs. We express some known constructions of MUBs in this form. Also it is shown how recently obtained results binding MUBs and some combinatorial structures (such as perfect nonlinear functions and relative difference sets) naturally follow from this criterion.Some directions for proving non-existence results are sketched as well.

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Section: Related Combinatorial Structuresmentioning

confidence: 99%

A Criterion for Attaining the Welch Bounds with Applications for Mutually Unbiased Bases

Belovs

Smotrovs

2008

Mathematical Methods in Computer Science

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“…[4,6,7,[9][10][11]13,14,16,17]). They can be used to construct DES-like cryptosystems that are resistant to differential attacks.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear functions and difference sets on group actions

Fan

2016

Des. Codes Cryptogr.

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Let G, H be finite groups and let X be a finite G-set. G-perfect nonlinear functions from X to H have been studied in several papers. They have more interesting properties than perfect nonlinear functions from G itself to H. By introducing the concept of a (G, H)-related difference family of X, we obtain a characterization of G-perfect nonlinear functions on X. When G is abelian, we characterize a G-difference set of X by the Fourier transform on a normalized G-dual set X. We will also investigate the existence and constructions of G-perfect nonlinear functions and G-bent functions. Several known results in [2,6,10,17] are direct consequences of our results.

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“…Furthermore, if ϕ fix {(0, y)|y ∈ H}, then we call ϕ an extended affine transformation (EA-transformation) of f , and we define EA(f ) = {g | ϕ(G f ) = G g , ϕ(0, H) = (0, H)}. It is pointed out that CCZ equivalence preserves linear and differential properties of functions for the case of abelian groups as well, see Remark 1 of [2], and [7], [10].…”

Section: Introductionmentioning

confidence: 99%