Information Technology Reference
In-Depth Information
Theorem 6 yields the complete characterization of PP of type
X
2
k
+
X
+
Tr
(
X
s
).
2
n
Corollary 2.
Let
1
≤
k
≤
n
−
1
and
1
≤
s
≤
−
2
.Then
X
2
k
+
X
+
Tr
(
X
s
)
is PP over
F
2
n
if and only if the following conditions are satisfied:
-
n
is odd
-
gcd
(
k, n
)=1
-
s
has binary weight 1 or 2.
Proof.
Firstly observe that the polynomial
X
2
k
+
X
has at least two zeros, 0 and
1. Hence from Claim 3 it follows that if
X
2
k
+
X
+
Tr
(
X
s
) is PP then necessarily
the mapping
L
(
x
)=
x
2
k
+
x
is 2- to -1. This holds if and only if gcd (
k, n
)=1.
Further note that the image set of such an
L
is the hyperplane
H
1
(0). Hence
γ
= 1 does not belong to the image set of
L
if and only if
Tr
(1) = 1, equivalently
if
n
is odd. The rest of the proof follows from Lemma 4 and Theorem 6 with
α
=
δ
=1and
β
=0.
Remark 2.
Some results of this paper are valid also in the finite fields of odd
characteristic. In a forthcoming paper we will report more accurately on that.
References
1. Bierbrauer, J., Kyureghyan, G.: Crooked binomials. Des. Codes Cryptogr. 46, 269-
301 (2008)
2. Dubuc, S.: Characterization of linear structures. Des. Codes Cryptogr. 22, 33-45
(2001)
3. Hollmann, H.D.L., Xing, Q.: A class of permutation polynomials of
F
2
n
related to
Dickson polynomials. Finite Fields Appl. 11(1), 111-122 (2005)
4. Kyureghyan, G.: Crooked maps in
F
2
n
. Finite Fields Appl. 13(3), 713-726 (2007)
5. Lai, X.: Additive and linear structures of cryptographic functions. In: Preneel, B.
(ed.) FSE 1994. LNCS, vol. 1008, pp. 75-85. Springer, Heidelberg (1995)
6. Laigle-Chapuy, Y.: A note on a class of quadratic permutations over
F
2
n
.In:Boztas,
S., Lu, H.-F(F.) (eds.) AAECC 2007. LNCS, vol. 4851, pp. 130-137. Springer, Hei-
delberg (2007)
7. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Ap-
plications 20
8. Yashchenko, V.V.: On the propagation criterion for Boolean functions and bent
functions. Problems of Information Transmission 33(1), 62-71 (1997)
9. Yuan, J., Ding, C., Wang, H., Pieprzyk, J.: Permutation polynomials of the form
(
x
p
−
x
+
δ
)
s
+
L
(
x
). Finite Fields Appl. 14(2), 482-493 (2008)
