On planar functions of elementary abelian $p$-group type

Minami, Kaori; Nakagawa, Nobuo

doi:10.14492/hokmj/1253539534

Cited by 5 publications

(4 citation statements)

References 19 publications

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“…[17]. The list of known commutative semifields is short [12,34,49], and Minami and Nakagawa [39] have determined the polynomial forms of certain commutative semifields. It may be of some interest to examine the known commutative semifields one by one to check whether their isotopes will yield planar functions of the form L(X) 2 − wX 2 , but the answer is most probably negative.…”

Section: A Function Approach To C-planes and H-planesmentioning

confidence: 99%

Finite flag-transitive affine planes with a solvable automorphism group

Feng

2017

Journal of Combinatorial Theory, Series A

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In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associating them with planar functions and permutation polynomials in the odd order and even order case respectively. In the odd order case, we characterize the Kantor-Suetake family by using Menichetti's classification of generalized twisted fields and Blokhuis, Lavrauw and Ball's classifcation of rank two commutative semifields. In the even order case, we develop a technique to study permutation polynomials of DO type by quadratic forms and characterize such planes that have dimensions up to four over their kernels. IntroductionLet V be a 2n-dimensional vector space over the finite field F q . A spread S of V is a collection of n-dimensional subspaces that partitions the nonzero vectors in V . The members of S are the components, and V is the ambient space. The kernel is the subring of ΓL(V ) that fixes each component, and it is a finite field containing F q . The dimension of S is the common value of the dimensions of its components over the kernel. The automorphism group Aut(S) is the subgroup of ΓL(V ) that maps components to components. The incidence structure Π(S) with point set V and line set {W + v : W ∈ S, v ∈ V } and incidence being inclusion is a translation plane. The kernel or dimension of Π(S) is that of S respectively. Andre [4] has shown that Aut(S) is the translation complement of the plane Π(S) and each finite translation plane can be obtained from a spread in this way. Two spreads S and S ′ of V are isomorphic if S ′ = {g(W ) : W ∈ S} for some g ∈ ΓL(V ), and isomorphic spreads correspond to isomorphic planes.An affine plane is called flag-transitive if it admits a collineation group which acts transitively on the flags, namely, the incident point-line pairs. Throughout this paper, we will only consider finite planes. Wagner [47] has shown that finite flag-transitive planes are necessarily translation planes, so the plane must have prime power order and can be constructed from a spread S with ambient space V of dimension 2n over F q for some n and q. The affine plane Π(S) constructed from a spread S is flag-transitive if and only if Aut(S) is transitive on the components. Foulser has determined all flag-transitive groups of finite affine planes in [24,25]. The only non-Desarguesian flag-transitive affine planes with nonsolvable collineation groups are the nearfield planes of order 9, the Hering plane of order 27 [23], and the Lüneburg planes of even order [36], cf. [13, 30]. In the solvable case, Foulser has shown that with a finite number of exceptions, which are explicitly described, a solvable flag transitive group of a finite affine plane is a subgroup of a one-dimensional Desarguesian affine plane.Kantor...

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Section: A Function Approach To C-planes and H-planesmentioning

confidence: 99%

Finite flag-transitive affine planes with a solvable automorphism group

Feng

2017

Journal of Combinatorial Theory, Series A

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“…The polynomial representations of functions (iii), (viii)-(x) can be found in [29]. Note that PN functions (3) of family (iv) and families (v) and (vi) were constructed by following patterns of some known families of APN functions over fields of even characteristic, see [5], [9].…”

Section: Known Cases Of Planar Functions and Commutative Semifieldsmentioning

confidence: 99%

Planar functions and commutative semifields

Budaghyan¹,

Helleseth²

2010

Tatra Mountains Mathematical Publications

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ABSTRACT. This paper gives a short survey on planar functions and commutative semifields and considers a possible extension of CCZ-equivalence which is the most general known equivalence relation of functions preserving the planar property. PN and APN functionsLet p be any prime number and n any positive integer. A function F from the field F p n to itself is called planar if all the equationshave exactly one solution, that is, if for any non-zero element a of Since 1991 planar functions have attracted interest also from cryptography as functions with optimal resistance to differential cryptanalysis. In this context they were first considered in the work of N y b e r g [32], where they were given a new name "perfect nonlinear" (PN) which described their important cryptographic property of being as far as possible from being linear (in certain sense). However, it is obvious that planar or PN functions exist only for p odd since if p is even and x 0 is a solution of (1)

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“…The representations of the Dickson PN functions can be found in [15]. The only known non-quadratic PN functions are the power functions x 3 t +1 2 over F 3 n , where t is odd and gcd(t, n) = 1 [7,13].…”

Section: Introductionmentioning

confidence: 99%

New commutative semifields defined by new PN multinomials

Budaghyan

Helleseth

2010

Cryptogr. Commun.

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We introduce two infinite classes of quadratic PN multinomials over F p 2k where p is any odd prime. We prove that for k odd one of these classes defines a new family of commutative semifields (in part by studying the nuclei of these semifields). After the works of Dickson (Trans Am Math Soc 7:514-522, 1906) and Albert (Trans Am Math Soc 72: [296][297][298][299][300][301][302][303][304][305][306][307][308][309] 1952), this is the firstly found infinite family of commutative semifields which is defined for all odd primes p. These results also imply that these PN functions are CCZ-inequivalent to all previously known PN mappings.

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On planar functions of elementary abelian $p$-group type

Cited by 5 publications

References 19 publications

Finite flag-transitive affine planes with a solvable automorphism group

Finite flag-transitive affine planes with a solvable automorphism group

Planar functions and commutative semifields

New commutative semifields defined by new PN multinomials

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