On alltop functions

Hamidli, Fuad
Let q be a power of an odd prime p and let Fq be a finite field. A map f is called planar on Fq if for any a ∈ F? q, the difference map (or derivative of f at a point a) Da(x) = f(x + a)− f(x) is bijective. The definition of Alltop function is that, the differencemapatpointainthegivenfieldofoddcharacteristicisitselfplanarforany a ∈ F∗ q. Alltop functions have special importance in cryptography and related areas. For example, they are used to construct mutually unbiased bases (MUB) in quantum information theory. The map x 7→ x3 is an Alltop function in all finite fields found by Alltop in 1980 which is an optimal function with respect to the known bounds on auto and crosscorrelation. Since then it was shown that these kind of functions do not exist when p = 3 (Hall, Rao, Donovan). So far, it has been found that xq+2 is also an Alltop function over finite field Fq2 where 3 does not divide q + 1 and this is EA-inequivalent to x3 whereas its difference function (derivative), which is planar, is EA-equivalent to x2 (Hall, Rao, Gagola). It is still an open problem whether there existanotherEA-inequivalentAlltopfunctionsoranymethodtoconstructnewAlltop functions. Inthisthesisclassificationofq-cubicAlltopbinomialsover Fq2 isgiven. Specifically, x3 + ux2q+1 in Fq2 for u ∈ F∗ q2 is analyzed and for this case permutation polynomials L1(x) = ax+bxq and L2(x) = cx+dxq arefoundthatsatisfy L1◦x3◦L2 = x3 +ux2q+1 and L1 ◦xq+2 ◦L2 = x3 + ux2q+1 for suitable values of u. Hence, by finding suitable values of u, it is shown that this class of functions are EA-equivalent to x3 and xq+2. Moreover, except x3 and the ones in its equivalence class, it is shown that there is no Alltop cubic q-monomials in Fq3. In addition, new notion ”p-ary Alltop functions" aredefinedfrom Fpn to Fp andtherelationbetweenAlltopfunctionsandp-aryAlltop functions over finite fields is given. Furthermore, some trivial and non-trivial p-ary Alltop functions are found and given.


Uniqueness of F-q-quadratic perfect nonlinear maps from F-q3 to F-q(2)
Özbudak, Ferruh (Elsevier BV, 2014-09-01)
Let q be a power of an odd prime. We prove that all F-q-quadratic perfect nonlinear maps from F-q3 to F-q(2) are equivalent. We also give a geometric method to find the corresponding equivalence explicitly.
Further results on rational points of the curve y(qn) - y = gamma xqh+1 - alpha over F-qm
Cosgun, Ayhan; Özbudak, Ferruh; SAYGI, ZÜLFÜKAR (2016-06-01)
Let q be a positive power of a prime number. For arbitrary positive integers h, n, m with n dividing m and arbitrary gamma, alpha is an element of F-qm with gamma not equal 0 the number of F-qm - rational points of the curve y(qn) - y = gamma x(qh+1) - alpha is determined in many cases (Ozbudak and Saygi, in: Larcher et al. (eds.) Applied algebra and number theory, 2014) with odd q. In this paper we complete some of the remaining cases for odd q and we also present analogous results for even q.
Some sufficient conditions for p-nilpotency of a finite group
Kızmaz, Muhammet Yasir (Informa UK Limited, 2019-09-02)
Let G be a finite group and let p be prime dividing . In this article, we supply some sufficient conditions for G to be p-nilpotent (see Theorem 1.2) as an extension of the main theorem of Li et al. (J. Group Theor. 20(1): 185-192, 2017).
Class groups of dihedral extensions
Lemmermeyer, F (Wiley, 2005-01-01)
Let L/F be a dihedral extension of degree 2p, where p is an odd prime. Let KIF and k/F be subextensions of L/F with degrees p and 2, respectively. Then we will study relations between the p-ranks of the class groups Cl(K) and Cl(k).
Invariant subspaces for banach space operators with a multiply connected spectrum
Yavuz, Onur (Springer Science and Business Media LLC, 2007-07-01)
We consider a multiply connected domain Omega = D \U (n)(j= 1) (B) over bar(lambda(j), r(j)) where D denotes the unit disk and (B) over bar(lambda(j), r(j)) subset of D denotes the closed disk centered at lambda(j) epsilon D with radius r(j) for j = 1,..., n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains delta Omega and does not contain the points lambda(1),lambda(2),...,lambda(n), and the operators T and r(j)( T -lambda I-j)(-1) are polynomially bounded, then th...
Citation Formats
F. Hamidli, “On alltop functions,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.