On alltop functions

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2017
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.
Citation Formats
F. Hamidli, “On alltop functions,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.