A class of functions and their application in constructing semi-biplanes and association schemes

Coulter, Robert S.; Henderson, Marie

doi:10.1016/s0012-365x(98)00345-8

Cited by 16 publications

(14 citation statements)

References 7 publications

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“…Almost perfect nonlinear functions have applications in cryptography [5,22] and coding theory [8]. They can also be used to construct association schemes and strongly regular graphs [32,33] and semibiplanes [10]. For more background and applications of almost perfect nonlinear functions, we recommend [20].…”

Section: Q and M Smallmentioning

confidence: 99%

Improved bounds on sizes of generalized caps in $AG(n,q)$

Tait

Won

2020

Preprint

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Section: Q and M Smallmentioning

confidence: 99%

Improved bounds on sizes of generalized caps in $AG(n,q)$

Tait

Won

2020

Preprint

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“…PN functions provide maximum resistance to differential attack. For odd p, PN (or, equivalently, differentially 1-uniform, or planar) functions exist, but for p = 2 they cannot, and the optimal functions in the even case are APN (differentially 2-uniform, or, in the terminology of Coulter and Henderson [8], semi-planar). Quadratic PN functions are equivalent to commutative presemifields (see [13, 9.3.2] or [9]).…”

Section: Generalized Ccz and Ea Classesmentioning

confidence: 99%

“…Quadratic PN functions are equivalent to commutative presemifields (see [13, 9.3.2] or [9]). APN functions define semibiplanes [8]. An APN function is also equivalent to a binary linear [2 n , 2 n −2n −1, 6] code, which is contained in the dual of the first-order Reed-Muller code (see [6]).…”

Section: Generalized Ccz and Ea Classesmentioning

confidence: 99%

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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“…For instance, they have been constructed in connection with several combinatorial and geometrical objects, such as semi-biplanes [14] and dual-hyperovals [16]. In this context these mappings are also called semi-planar [15].…”

Section: Introductionmentioning

confidence: 99%

On the infiniteness of a family of APN functions

Bartoli¹,

Calderini²,

Polverino³

et al. 2021

Preprint

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APN functions play a fundamental role in cryptography against attacks on block ciphers. Several families of quadratic APN functions have been proposed in the recent years, whose construction relies on the existence of specific families of polynomials.A key question connected with such constructions is to determine whether such APN functions exist for infinitely many dimensions or not.In this paper we consider a family of functions recently introduced by Li et al. in 2021 showing that for any dimension m ≥ 3 there exists an APN function belonging to such a family.Our main result is proved by a combination of different techniques arising from both algebraic varieties over finite fields connected with linearized permutation rational functions and partial vector space partitions, together with investigations on the kernels of linearized polynomials.

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A class of functions and their application in constructing semi-biplanes and association schemes

Cited by 16 publications

References 7 publications

Improved bounds on sizes of generalized caps in $AG(n,q)$

Improved bounds on sizes of generalized caps in $AG(n,q)$

Relative difference sets, graphs and inequivalence of functions between groups

On the infiniteness of a family of APN functions

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