Lilya Budaghyan

New classes of almost bent and almost perfect nonlinear polynomials

New infinite classes of almost bent and almost perfect nonlinear polynomials are constructed. It is shown that they are affine inequivalent to any sum of a power function and an affine function

Two Classes of Quadratic APN Binomials Inequivalent to Power Functions

This paper introduces the first found infinite classes of almost perfect nonlinear (APN) polynomials which are not Carlet-Charpin-Zinoviev (CCZ)-equivalent to power functions (at least for some values of the number of variables). These are two classes of APN binomials from F2n to F2n (for n divisible by 3, resp., 4). We prove that these functions are extended affine (EA)-inequivalent to any power function and that they are CCZ-inequivalent to the Gold, Kasami, inverse, and Dobbertin functions when n ges 12. This means that for n even they are CCZ-inequivalent to any known APN function. In particular, for n = 12,20,24, they are therefore CCZ-inequivalent to any power function.

Classes of Quadratic APN Trinomials and Hexanomials and Related Structures

A method for constructing differentially 4-uniform quadratic hexanomials has been recently introduced by J. Dillon. We give various generalizations of this method and we deduce the constructions of new infinite classes of almost perfect nonlinear quadratic trinomials and hexanomials from F22m to F22m. We check for m = 3 that some of these functions are CCZ-inequivalent to power functions.

An infinite class of quadratic APN functions which are not equivalent to power mappings

We exhibit an infinite class of almost perfect nonlinear quadratic polynomials from F2n to F2n (n ges 12, n divisible by 3 but not by 9). We prove that these functions are EA-inequivalent to any power function and that they are CCZ-inequivalent to any Gold function. In a forthcoming full paper, we shall also prove that at least some of these functions are CCZ-inequivalent to any Kasami function

Further Results on Niho Bent Functions

This paper consists of two main contributions. First, the Niho bent function consisting of 2r exponents (discovered by Leander and Kholosha) is studied. The dual of the function is found and it is shown that this new bent function is not of the Niho type. Second, all known univariate representations of Niho bent functions are analyzed for their relation to the completed Maiorana-McFarland class M. In particular, it is proven that two families do not belong to the completed class M. The latter result gives a positive answer to an open problem whether the class H of bent functions introduced by Dillon in his thesis of 1974 differs from the completed class M.

New Perfect Nonlinear Multinomials over F

We introduce two infinite families of perfect nonlinear Dembowski-Ostrom multinomials over

_{p^{2k}}

\textbf{F}_{p^{2k}}

for Any Odd Prime p

where pis any odd prime. We prove that in general these functions are CCZ-inequivalent to previously known PN mappings. One of these families has been constructed by extension of a known family of APN functions over

New commutative semifields defined by new PN multinomials

\textbf{F}_{2^{2k}}

The Simplest Method for Constructing APN Polynomials EA-Inequivalent to Power Functions

. This shows that known classes of APN functions over fields of even characteristic can serve as a source for further constructions of PN mappings over fields of odd characteristics. Besides, we supply results indicating that these PN functions define new commutative semifields. After the works of Dickson (1906) and Albert (1952), these are the firstly found infinite families of commutative semifields which are defined for all odd primes p.