17 research outputs found
Crooked maps in F2n
AbstractA map f:F2n→F2n is called crooked if the set {f(x+a)+f(x):x∈F2n} is an affine hyperplane for every fixed a∈F2n∗ (where F2n is considered as a vector space over F2). We prove that the only crooked power maps are the quadratic maps x2i+2j with gcd(n,i−j)=1. This is a consequence of the following result of independent interest: for any prime p and almost all exponents 0⩽d⩽pn−2 the set {xd+γ(x+a)d:x∈Fpn} contains n linearly independent elements, where γ and a≠0 are arbitrary elements from Fpn
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Coding Theory
Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances
Image sets of perfectly nonlinear maps
We present a lower bound on the image size of a -uniform map, ,
of finite fields, by extending the methods used for planar maps. In the
particularly interesting case of APN maps on binary fields, our bound coincides
with the one obtained by Ingo Czerwinski, using a linear programming method.
We study properties of APN maps of with minimal image set.
In particular, we observe that for even , a Dembowski-Ostrom polynomial of
form is APN if and only if is almost-3-to-1, that is when
its image set is minimal. We show that any almost-3-to-1 quadratic map is APN,
if is even. For odd, we present APN Dembowski-Ostrom polynomials on
with image sizes and .
We present several results connecting the image sets of special APN maps with
their Walsh spectrum. Especially, we show that a large class of APN maps has
the classical Walsh spectrum. Finally, we prove that the image size of a
non-bijective almost bent map contains at most elements.Comment: Minor revision with new references; Theorems 18, 19 are adde
Crooked functions
International audienceCrooked permutations were introduced twenty years ago since they allow to construct interesting objects in graph theory. The field of applications was extended later. Crooked functions, bijective or not, correspond to APN functions and to some optimal codes. We adopt an unified presentation of crooked functions, explaining the connection with partially-bent functions. We then complete some known results and derive new properties. For instance, we observe that crooked functions allow to construct sets of bent functions and define some permutations
Crooked functions
Crooked permutations were defined twenty years ago. It was firstly shown that they can be used to construct interesting objects in graph theory. The field of applications was extended later, since crooked functions, bijective or not, correspond to APN functions and to some optimal codes. We adopt an unified presentation, of crooked functions, explaining the connexion with partially-bent functions. We then complete some known results and propose new properties. For instance, crooked functions allow to construct sets of bent functions, or simply define some permutations
On a remarkable property of APN Gold functions
In [13] for a given vectorial Boolean function from to itself it was defined an associated Boolean function in variables that takes value~ iff and equation has solutions. In this paper we introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. It is an interesting open problem to describe differential equivalence class of a given APN function.
We consider the APN Gold function , where gcd, and prove that there exist exactly distinct affine functions such that and are differentially equivalent if for some and ; otherwise the number of such affine functions is equal to . This theoretical result and computer calculations obtained show that APN Gold functions for and are the only functions (except one function in 6 variables) among all known quadratic APN functions in variables that have more than trivial affine functions , where , preserving the associated Boolean function when adding to
The crooked property
International audienceCrooked permutations were introduced twenty years ago to cons- truct interesting objects in graph theory. These functions, over F2n with odd , are such that their derivatives have as image set a com- plement of a hyperplane. The field of applications was extended later, in particular to cryptography. However binary crooked functions are rare. It is still unknown if non quadratic crooked functions do ex- ist. We extend the concept and propose to study the crooked property for any characteristic. A function , from Fpn to itself, satisfies this property if all its derivatives have as image set an a ne subspace. We show that the partially-bent vectorial functions and the functions satisfying the crooked property are strongly related. We later focus on the components of these functions, establishing that the existence of linear structures is here decisive. We then propose a symbolic ap- proach to identify the linear structures. We claim that this problem consists in solving a system of linear equations, and can often be seen as a combinatorial problem