A map
f
:
F
2
n
→
F
2
n
is called crooked if the set
{
f
(
x
+
a
)
+
f
(
x
)
:
x
∈
F
2
n
}
is the complement of a hyperplane for every fixed
a
∈
F
2
n
∗
(where
F
2
n
is considered as a vector space ...over
F
2
). We prove that the only crooked power maps are the quadratic maps
x
2
k
+
2
l
with
gcd
(
n
,
k
−
l
)
=
1
.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Based on a computer search, Anne Canteaut conjectured that the exponent $2^{2r}+2^r+1$ in ${\bf F}_{2^{6r}}$ and the exponent $(2^{r}+1)^2$ in ${\bf F}_{2^{4r}}$ yield bent monomial functions. These ...conjectures are proved in A. Canteaut, P. Charpin, and G. Kyureghyan, A new class of monomial bent functions, in Proceedings of the 2006 IEEE International Symposium on Information Theory, (ISIT 06 Seattle), IEEE Press, Piscataway, NJ, 2006, pp. 903-906 and N. G. Leander, IEEE Trans. Inform. Theory, 52 (2006), pp. 738-743. Both exponents are of binary weight $3$ and define functions from the Maiorana-McFarland class $\mathcal{M}$ of bent functions to the subfield. In this paper we show that these are the only such exponents. Our proof is based on the classification of the permutation binomials $X^{2^k+2}+ \nu X$ of finite fields of even characteristics. We also extend the result of Leander, determining all bent monomial functions with the exponent $(2^{r}+1)^2$.
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CEKLJ, DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
43.
Crooked maps in F 2 n Kyureghyan, Gohar M.
Finite fields and their applications,
07/2007, Volume:
13, Issue:
3
Journal Article
Peer reviewed
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We consider image sets of differentially \(d\)-uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution, by extending methods used ...for planar maps. We apply the results to study \(d\)-uniform Dembowski-Ostrom polynomials. Further, we focus on a particularly interesting case of APN maps on binary fields. We show that APN maps with the minimal image size must have a very special preimage distribution. We prove that for an even \(n\) the image sets of several well-studied families of APN maps are minimal. We present results connecting the image sets of special maps with their Walsh spectrum. Especially, we show that the fact that several large classes of APN maps have the classical Walsh spectrum is explained by the minimality of their image sets. Finally, we present upper bounds on the image size of APN maps.
In this paper we consider in detail the composition of an irreducible polynomial with X^2 and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd ...characteristics. More precisely, given an irreducible polynomial of degree n and order 2^rt with t odd, the construction produces ord_t(2)/d irreducible polynomials of degree n and order t for a certain divisor d of n. The construction can be used, for example, to search irreducible polynomials with specific requirements on its coefficients.
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