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Since
k
(
k−
1
2
=2
k
=10for
m
= 4, there exist some IDS3-to-1 APN functions
and experimentally we found some of them. In the remaining part of this section
we will show that some known important APN functions are S3-to-1 functions.
1. Power APN Functions
The power APN functions on even number of variables are of the form
F
(
X
)=
X
3
d
where gcd(
d, k
)=1and
k
=
2
m
−
1
3
.Let
α
be a primitive element of
V
m
.Since
m
is even,
V
2
is a subfield of
V
m
with
β
=
α
k
a generator of
V
2
1
,β,β
2
=
{
}
.
Now consider
P
i
+1
=
α
i
V
2
α
i
,α
i
β, α
i
β
2
=
{
}
for 0
≤
i<k
and
P
0
=
{
0
}
.
makes a disjoint partition over
V
m
and
α
i
+
α
i
β
+
α
i
β
2
{
P
i
,
0
≤
i
≤
k
}
=
α
i
(1 +
β
+
β
2
)=
α
i
.
0=0for0
i<k
. Now for the S-box
F
(
X
)=
X
3
d
,
≤
α
3
di
,α
3
di
β
3
d
,α
3
di
β
6
d
=
α
3
di
F
(
P
i
+1
)=
{
}
for 0
≤
i<k
.Sincegcd(
d, k
)=1,
}
{
α
3
di
=
α
3
dj
F
−
1
(
α
3
di
)
α
3
i
,
0
for 0
≤
i<j<k
i.e.,
|
|
=3.Here
U
=
{
0
≤
. Therefore, the APN power functions i.e.,
X
3
d
,
gcd(
d, k
)=1satisfies
Construction 1. Hence, the power APN functions follow the restriction imposed
on a S-box to be APN in Theorem 2.
2.
F
(
X
)=
X
3
+
tr
(
X
9
)
The function
F
(
X
)=
X
3
+
tr
(
X
9
)(where
tr
(
X
)=
m−
1
i
=0
i
≤
k
−
1
}
X
2
i
is the trace
function from
V
m
to
V
1
) is APN function and when
m
≥
7and
m>
2
p
where
p
is the smallest positive integer such that
m
=3andgcd(
m, p
)=1,
X
3
+
tr
(
X
9
) is CCZ-inequivalent to all power functions on
V
m
[3]. Similar to the
power function case (i.e., Item 1), one can easily prove that
F
(
α
i
)=
F
(
α
i
+
k
)=
F
(
α
i
+2
k
)for0
=1
,m
2
m
−
1
3
≤
i<k
=
and
F
(0) = 0 where
α
is a primitive element
in
V
m
.Let
x
=
α
i
and
y
=
α
j
where 0
≤
i<j<k
. Now we will show
that
F
(
x
)
=
F
(
y
) i.e.,
x
3
+
tr
(
x
9
)
=
y
3
+
tr
(
y
9
) i.e.,
tr
(
x
9
+
y
9
)
=
x
3
+
y
3
.If
tr
(
x
9
+
y
9
) = 0 then we are done because
x
3
=
y
3
. Now consider
tr
(
x
9
+
y
9
)=1.If
x
3
+
y
3
= 1 i.e.,
y
3
=1+
x
3
then
tr
(
x
9
+
y
9
)=
tr
(
x
9
+(1+
x
3
)
3
)=
tr
(1+
x
3
+
x
6
)=
m−
1
i
=0
(1 +
x
3
+
x
6
)
2
i
=
m−
1
i
=0
(1 +
x
3
.
2
i
+
x
6
.
2
i
)=
m−
1
i
=0
(1 +
x
3
.
2
i
+
x
3
.
2
i
+1
)=
x
3
+
x
3
.
2
m
=
x
3
+
x
3
=0whichisacontradiction. Therefore
F
(
x
)
=
F
(
y
) implies
X
3
+
tr
(
X
9
) is S3-to-1 function.
The general study of APN property of S3-to-1 functions can give clearer pic-
ture to generalize the APN power functions on even number of variables and
constructions of new class of APN functions. Overall the S3-to-1 functions cov-
ers many interesting parts of the studies of APN functions. The study on finding
the exact relation of the ordered set
U
and the partitions
P
i
(or, flats
P
i
∪
P
0
)
which makes
F
APN will be very interesting.
4 Power Function
In this section we present a necessary condition for a power function,
F
:
X
→
X
d
,X
V
m
, to be APN. Unlike the previous section, in this section we study for
general
m
unless it is specified as even or odd. The complete characterization of
all APN power functions is not known. Some results on the necessary conditions
for a power functionto be APN are available in recent literature. If
F
is APN
then gcd(
d,
2
m
∈
1) = 1 for odd
m
and gcd(
d,
2
m
−
−
1) = 3 for even
m
[1]. For
